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# Dense matrices using a NumPy backend.

Dense matrices using a NumPy backend.

This serves as a base class for dense matrices over Real Double Field and Complex Double Field.

AUTHORS:

- Jason Grout, Sep 2008: switch to NumPy backend, factored out the Matrix_double_dense class
- Josh Kantor
- William Stein: many bug fixes and touch ups.

EXAMPLES:

    sage: b=Mat(RDF,2,3).basis()
    sage: b[0]
    [1.0 0.0 0.0]
    [0.0 0.0 0.0]

We deal with the case of zero rows or zero columns:

    sage: m = MatrixSpace(RDF,0,3)
    sage: m.zero_matrix()
    []

TESTS:

    sage: a = matrix(RDF,2,range(4), sparse=False)
    sage: TestSuite(a).run()
    sage: a = matrix(CDF,2,range(4), sparse=False)
    sage: TestSuite(a).run()

class `sage.matrix.matrix_double_dense.``Matrix_double_dense`

  Bases: `sage.matrix.matrix_dense.Matrix_dense`

  Base class for matrices over the Real Double Field and the Complex Double Field. These are supposed to be fast matrix operations using C doubles. Most operations are implemented using numpy which will call the underlying BLAS on the system.

  This class cannot be instantiated on its own. The numpy matrix creation depends on several variables that are set in the subclasses.

  EXAMPLES:

      sage: m = Matrix(RDF, [[1,2],[3,4]])
      sage: m**2
      [ 7.0 10.0]
      [15.0 22.0]
      sage: m^(-1)        # rel tol 1e-15
      [-1.9999999999999996  0.9999999999999998]
      [ 1.4999999999999998 -0.4999999999999999]

  `LU`()

    Returns a decomposition of the (row-permuted) matrix as a product of a lower-triangular matrix (“L”) and an upper-triangular matrix (“U”).

    OUTPUT:

    For an \(m\times n\) matrix `A` this method returns a triple of immutable matrices `P, L, U` such that

    - `P*A = L*U`
    - `P` is a square permutation matrix, of size \(m\times m\), so is all zeroes, but with exactly a single one in each row and each column.
    - `L` is lower-triangular, square of size \(m\times m\), with every diagonal entry equal to one.
    - `U` is upper-triangular with size \(m\times n\), i.e. entries below the “diagonal” are all zero.

    The computed decomposition is cached and returned on subsequent calls, thus requiring the results to be immutable.

    Effectively, `P` permutes the rows of `A`. Then `L` can be viewed as a sequence of row operations on this matrix, where each operation is adding a multiple of a row to a subsequent row. There is no scaling (thus 1’s on the diagonal of `L`) and no row-swapping (`P` does that). As a result `U` is close to being the result of Gaussian-elimination. However, round-off errors can make it hard to determine the zero entries of `U`.

    Note

    Sometimes this decomposition is written as `A=P*L*U`, where `P` represents the inverse permutation and is the matrix inverse of the `P` returned by this method. The computation of this matrix inverse can be accomplished quickly with just a transpose as the matrix is orthogonal/unitary.

    EXAMPLES:

        sage: m = matrix(RDF,4,range(16))
        sage: P,L,U = m.LU()
        sage: P*m
        [12.0 13.0 14.0 15.0]
        [ 0.0  1.0  2.0  3.0]
        [ 8.0  9.0 10.0 11.0]
        [ 4.0  5.0  6.0  7.0]
        sage: L*U # rel tol 2e-16
        [12.0 13.0 14.0 15.0]
        [ 0.0  1.0  2.0  3.0]
        [ 8.0  9.0 10.0 11.0]
        [ 4.0  5.0  6.0  7.0]

    Trac 10839 made this routine available for rectangular matrices.

        sage: A = matrix(RDF, 5, 6, range(30)); A
        [ 0.0  1.0  2.0  3.0  4.0  5.0]
        [ 6.0  7.0  8.0  9.0 10.0 11.0]
        [12.0 13.0 14.0 15.0 16.0 17.0]
        [18.0 19.0 20.0 21.0 22.0 23.0]
        [24.0 25.0 26.0 27.0 28.0 29.0]
        sage: P, L, U = A.LU()
        sage: P
        [0.0 0.0 0.0 0.0 1.0]
        [1.0 0.0 0.0 0.0 0.0]
        [0.0 0.0 1.0 0.0 0.0]
        [0.0 0.0 0.0 1.0 0.0]
        [0.0 1.0 0.0 0.0 0.0]
        sage: L.zero_at(0)   # Use zero_at(0) to get rid of signed zeros
        [ 1.0  0.0  0.0  0.0  0.0]
        [ 0.0  1.0  0.0  0.0  0.0]
        [ 0.5  0.5  1.0  0.0  0.0]
        [0.75 0.25  0.0  1.0  0.0]
        [0.25 0.75  0.0  0.0  1.0]
        sage: U.zero_at(0)   # Use zero_at(0) to get rid of signed zeros
        [24.0 25.0 26.0 27.0 28.0 29.0]
        [ 0.0  1.0  2.0  3.0  4.0  5.0]
        [ 0.0  0.0  0.0  0.0  0.0  0.0]
        [ 0.0  0.0  0.0  0.0  0.0  0.0]
        [ 0.0  0.0  0.0  0.0  0.0  0.0]
        sage: P*A-L*U
        [0.0 0.0 0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0 0.0 0.0]
        sage: P.transpose()*L*U
        [ 0.0  1.0  2.0  3.0  4.0  5.0]
        [ 6.0  7.0  8.0  9.0 10.0 11.0]
        [12.0 13.0 14.0 15.0 16.0 17.0]
        [18.0 19.0 20.0 21.0 22.0 23.0]
        [24.0 25.0 26.0 27.0 28.0 29.0]

    Trivial cases return matrices of the right size and characteristics.

        sage: A = matrix(RDF, 5, 0, entries=0)
        sage: P, L, U = A.LU()
        sage: P.parent()
        Full MatrixSpace of 5 by 5 dense matrices over Real Double Field
        sage: L.parent()
        Full MatrixSpace of 5 by 5 dense matrices over Real Double Field
        sage: U.parent()
        Full MatrixSpace of 5 by 0 dense matrices over Real Double Field
        sage: P*A-L*U
        []

    The results are immutable since they are cached.

        sage: P, L, U = matrix(RDF, 2, 2, range(4)).LU()
        sage: L[0,0] = 0
        Traceback (most recent call last):
            ...
        ValueError: matrix is immutable; please change a copy instead (i.e., use copy(M) to change a copy of M).
        sage: P[0,0] = 0
        Traceback (most recent call last):
            ...
        ValueError: matrix is immutable; please change a copy instead (i.e., use copy(M) to change a copy of M).
        sage: U[0,0] = 0
        Traceback (most recent call last):
            ...
        ValueError: matrix is immutable; please change a copy instead (i.e., use copy(M) to change a copy of M).

  `LU_valid`()

    Returns `True` if the LU form of this matrix has already been computed.

    EXAMPLES:

        sage: A = random_matrix(RDF,3) ; A.LU_valid()
        False
        sage: P, L, U = A.LU()
        sage: A.LU_valid()
        True

  `QR`()

    Returns a factorization into a unitary matrix and an upper-triangular matrix.

    INPUT:

    Any matrix over `RDF` or `CDF`.

    OUTPUT:

    `Q`, `R` – a pair of matrices such that if \(A\) is the original matrix, then

    \[A = QR, \quad Q^\ast Q = I\]

    where \(R\) is upper-triangular. \(Q^\ast\) is the conjugate-transpose in the complex case, and just the transpose in the real case. So \(Q\) is a unitary matrix (or rather, orthogonal, in the real case), or equivalently \(Q\) has orthogonal columns. For a matrix of full rank this factorization is unique up to adjustments via multiples of rows and columns by multiples with scalars having modulus \(1\). So in the full-rank case, \(R\) is unique if the diagonal entries are required to be positive real numbers.

    The resulting decomposition is cached.

    ALGORITHM:

    Calls “linalg.qr” from SciPy, which is in turn an interface to LAPACK routines.

    EXAMPLES:

    Over the reals, the inverse of `Q` is its transpose, since including a conjugate has no effect. In the real case, we say `Q` is orthogonal.

        sage: A = matrix(RDF, [[-2, 0, -4, -1, -1],
        ....:                  [-2, 1, -6, -3, -1],
        ....:                  [1, 1, 7, 4, 5],
        ....:                  [3, 0, 8, 3, 3],
        ....:                  [-1, 1, -6, -6, 5]])
        sage: Q, R = A.QR()

    At this point, `Q` is only well-defined up to the signs of its columns, and similarly for `R` and its rows, so we normalize them:

        sage: Qnorm = Q._normalize_columns()
        sage: Rnorm = R._normalize_rows()
        sage: Qnorm.round(6).zero_at(10^-6)
        [ 0.458831  0.126051  0.381212  0.394574   0.68744]
        [ 0.458831  -0.47269 -0.051983 -0.717294  0.220963]
        [-0.229416 -0.661766  0.661923  0.180872 -0.196411]
        [-0.688247 -0.189076 -0.204468  -0.09663  0.662889]
        [ 0.229416 -0.535715 -0.609939  0.536422 -0.024551]
        sage: Rnorm.round(6).zero_at(10^-6)
        [ 4.358899 -0.458831 13.076697  6.194225  2.982405]
        [      0.0  1.670172  0.598741  -1.29202  6.207997]
        [      0.0       0.0  5.444402  5.468661 -0.682716]
        [      0.0       0.0       0.0  1.027626   -3.6193]
        [      0.0       0.0       0.0       0.0  0.024551]
        sage: (Q*Q.transpose())  # tol 1e-14
        [0.9999999999999994                0.0                0.0                0.0                0.0]
        [               0.0                1.0                0.0                0.0                0.0]
        [               0.0                0.0 0.9999999999999999                0.0                0.0]
        [               0.0                0.0                0.0 0.9999999999999998                0.0]
        [               0.0                0.0                0.0                0.0 1.0000000000000002]
        sage: (Q*R - A).zero_at(10^-14)
        [0.0 0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0 0.0]

    Now over the complex numbers, demonstrating that the SciPy libraries are (properly) using the Hermitian inner product, so that `Q` is a unitary matrix (its inverse is the conjugate-transpose).

        sage: A = matrix(CDF, [[-8, 4*I + 1, -I + 2, 2*I + 1],
        ....:                  [1, -2*I - 1, -I + 3, -I + 1],
        ....:                  [I + 7, 2*I + 1, -2*I + 7, -I + 1],
        ....:                  [I + 2, 0, I + 12, -1]])
        sage: Q, R = A.QR()
        sage: Q._normalize_columns()  # tol 1e-6
        [                           0.7302967433402214    0.20705664550556482 + 0.5383472783144685*I   0.24630498099986423 - 0.07644563587232917*I   0.23816176831943323 - 0.10365960327796941*I]
        [                         -0.09128709291752768  -0.20705664550556482 - 0.37787837804765584*I   0.37865595338630315 - 0.19522214955246678*I    0.7012444502144682 - 0.36437116509865947*I]
        [  -0.6390096504226938 - 0.09128709291752768*I    0.17082173254209104 + 0.6677576817554466*I -0.03411475806452064 + 0.040901987417671426*I   0.31401710855067644 - 0.08251917187054114*I]
        [ -0.18257418583505536 - 0.09128709291752768*I  -0.03623491296347384 + 0.07246982592694771*I    0.8632284069415112 + 0.06322839976356195*I  -0.44996948676115206 - 0.01161191812089182*I]
        sage: R._normalize_rows().zero_at(1e-15)  # tol 1e-6
        [                        10.954451150103322                      -1.9170289512680814*I   5.385938482134133 - 2.1908902300206643*I -0.2738612787525829 - 2.1908902300206643*I]
        [                                       0.0                            4.8295962564173  -0.8696379111233719 - 5.864879483945123*I  0.993871898426711 - 0.30540855212070794*I]
        [                                       0.0                                        0.0                          12.00160760935814 -0.2709533402297273 + 0.4420629644486325*I]
        [                                       0.0                                        0.0                                        0.0                         1.9429639442589917]
        sage: (Q.conjugate().transpose()*Q).zero_at(1e-15)  # tol 1e-15
        [               1.0                0.0                0.0                0.0]
        [               0.0 0.9999999999999994                0.0                0.0]
        [               0.0                0.0 1.0000000000000002                0.0]
        [               0.0                0.0                0.0 1.0000000000000004]
        sage: (Q*R - A).zero_at(10^-14)
        [0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0]

    An example of a rectangular matrix that is also rank-deficient. If you run this example yourself, you may see a very small, nonzero entries in the third row, in the third column, even though the exact version of the matrix has rank 2. The final two columns of `Q` span the left kernel of `A` (as evidenced by the two zero rows of `R`). Different platforms will compute different bases for this left kernel, so we do not exhibit the actual matrix.

        sage: Arat = matrix(QQ, [[2, -3, 3],
        ....:                    [-1, 1, -1],
        ....:                    [-1, 3, -3],
        ....:                    [-5, 1, -1]])
        sage: Arat.rank()
        2
        sage: A = Arat.change_ring(CDF)
        sage: Q, R = A.QR()
        sage: R._normalize_rows()  # abs tol 1e-14
        [     5.567764362830022    -2.6940795304016243     2.6940795304016243]
        [                   0.0     3.5695847775155825    -3.5695847775155825]
        [                   0.0                    0.0 2.4444034681064287e-16]
        [                   0.0                    0.0                    0.0]
        sage: (Q.conjugate_transpose()*Q)  # abs tol 1e-14
        [     1.0000000000000002  -5.185196889911925e-17 -4.1457180570414476e-17  -2.909388767229071e-17]
        [ -5.185196889911925e-17      1.0000000000000002  -9.286869233696149e-17 -1.1035822863186828e-16]
        [-4.1457180570414476e-17  -9.286869233696149e-17                     1.0  4.4159215672155694e-17]
        [ -2.909388767229071e-17 -1.1035822863186828e-16  4.4159215672155694e-17                     1.0]

    Results are cached, meaning they are immutable matrices. Make a copy if you need to manipulate a result.

        sage: A = random_matrix(CDF, 2, 2)
        sage: Q, R = A.QR()
        sage: Q.is_mutable()
        False
        sage: R.is_mutable()
        False
        sage: Q[0,0] = 0
        Traceback (most recent call last):
        ...
        ValueError: matrix is immutable; please change a copy instead (i.e., use copy(M) to change a copy of M).
        sage: Qcopy = copy(Q)
        sage: Qcopy[0,0] = 679
        sage: Qcopy[0,0]
        679.0

    TESTS:

    Trivial cases return trivial results of the correct size, and we check `Q` itself in one case, verifying a fix for trac ticket #10795.

        sage: A = zero_matrix(RDF, 0, 10)
        sage: Q, R = A.QR()
        sage: Q.nrows(), Q.ncols()
        (0, 0)
        sage: R.nrows(), R.ncols()
        (0, 10)
        sage: A = zero_matrix(RDF, 3, 0)
        sage: Q, R = A.QR()
        sage: Q.nrows(), Q.ncols()
        (3, 3)
        sage: R.nrows(), R.ncols()
        (3, 0)
        sage: Q
        [1.0 0.0 0.0]
        [0.0 1.0 0.0]
        [0.0 0.0 1.0]

  `SVD`()

    Return the singular value decomposition of this matrix.

    The U and V matrices are not unique and may be returned with different values in the future or on different systems. The S matrix is unique and contains the singular values in descending order.

    The computed decomposition is cached and returned on subsequent calls.

    INPUT:

    - A – a matrix

    OUTPUT:

    - U, S, V – immutable matrices such that \(A = U*S*V.conj().transpose()\) where U and V are orthogonal and S is zero off of the diagonal.

    Note that if self is m-by-n, then the dimensions of the matrices that this returns are (m,m), (m,n), and (n, n).

    Note

    If all you need is the singular values of the matrix, see the more convenient `singular_values()`.

    EXAMPLES:

        sage: m = matrix(RDF,4,range(1,17))
        sage: U,S,V = m.SVD()
        sage: U*S*V.transpose()  # tol 1e-14
        [0.9999999999999993 1.9999999999999987  3.000000000000001  4.000000000000002]
        [ 4.999999999999998  5.999999999999998  6.999999999999998                8.0]
        [ 8.999999999999998  9.999999999999996 10.999999999999998               12.0]
        [12.999999999999998               14.0               15.0               16.0]

    A non-square example:

        sage: m = matrix(RDF, 2, range(1,7)); m
        [1.0 2.0 3.0]
        [4.0 5.0 6.0]
        sage: U, S, V = m.SVD()
        sage: U*S*V.transpose()  # tol 1e-14
        [0.9999999999999994 1.9999999999999998  2.999999999999999]
        [ 4.000000000000001  5.000000000000002  6.000000000000001]

    S contains the singular values:

        sage: S.round(4)
        [ 9.508    0.0    0.0]
        [   0.0 0.7729    0.0]
        sage: [round(sqrt(abs(x)),4) for x in (S*S.transpose()).eigenvalues()]
        [9.508, 0.7729]

    U and V are orthogonal matrices:

        sage: U # random, SVD is not unique
        [-0.386317703119 -0.922365780077]
        [-0.922365780077  0.386317703119]
        [-0.274721127897 -0.961523947641]
        [-0.961523947641  0.274721127897]
        sage: (U*U.transpose())  # tol 1e-15
        [               1.0                0.0]
        [               0.0 1.0000000000000004]
        sage: V # random, SVD is not unique
        [-0.428667133549  0.805963908589  0.408248290464]
        [-0.566306918848  0.112382414097 -0.816496580928]
        [-0.703946704147 -0.581199080396  0.408248290464]
        sage: (V*V.transpose())  # tol 1e-15
        [0.9999999999999999                0.0                0.0]
        [               0.0                1.0                0.0]
        [               0.0                0.0 0.9999999999999999]

    TESTS:

        sage: m = matrix(RDF,3,2,range(1, 7)); m
        [1.0 2.0]
        [3.0 4.0]
        [5.0 6.0]
        sage: U,S,V = m.SVD()
        sage: U*S*V.transpose()  # tol 1e-15
        [0.9999999999999996 1.9999999999999998]
        [               3.0 3.9999999999999996]
        [ 4.999999999999999  6.000000000000001]

        sage: m = matrix(RDF, 3, 0, []); m
        []
        sage: m.SVD()
        ([], [], [])
        sage: m = matrix(RDF, 0, 3, []); m
        []
        sage: m.SVD()
        ([], [], [])
        sage: def shape(x): return (x.nrows(), x.ncols())
        sage: m = matrix(RDF, 2, 3, range(6))
        sage: map(shape, m.SVD())
        [(2, 2), (2, 3), (3, 3)]
        sage: for x in m.SVD(): x.is_immutable()
        True
        True
        True

  `cholesky`()

    Returns the Cholesky factorization of a matrix that is real symmetric, or complex Hermitian.

    INPUT:

    Any square matrix with entries from `RDF` that is symmetric, or with entries from `CDF` that is Hermitian. The matrix must be positive definite for the Cholesky decomposition to exist.

    OUTPUT:

    For a matrix \(A\) the routine returns a lower triangular matrix \(L\) such that,

    \[A = LL^\ast\]

    where \(L^\ast\) is the conjugate-transpose in the complex case, and just the transpose in the real case. If the matrix fails to be positive definite (perhaps because it is not symmetric or Hermitian), then this function raises a `ValueError`.

    IMPLEMENTATION:

    The existence of a Cholesky decomposition and the positive definite property are equivalent. So this method and the `is_positive_definite()` method compute and cache both the Cholesky decomposition and the positive-definiteness. So the `is_positive_definite()` method or catching a `ValueError` from the `cholesky()` method are equally expensive computationally and if the decomposition exists, it is cached as a side-effect of either routine.

    EXAMPLES:

    A real matrix that is symmetric and positive definite.

        sage: M = matrix(RDF,[[ 1,  1,    1,     1,     1],
        ....:                 [ 1,  5,   31,   121,   341],
        ....:                 [ 1, 31,  341,  1555,  4681],
        ....:                 [ 1,121, 1555,  7381, 22621],
        ....:                 [ 1,341, 4681, 22621, 69905]])
        sage: M.is_symmetric()
        True
        sage: L = M.cholesky()
        sage: L.round(6).zero_at(10^-10)
        [   1.0    0.0         0.0        0.0     0.0]
        [   1.0    2.0         0.0        0.0     0.0]
        [   1.0   15.0   10.723805        0.0     0.0]
        [   1.0   60.0   60.985814   7.792973     0.0]
        [   1.0  170.0  198.623524  39.366567  1.7231]
        sage: (L*L.transpose()).round(6).zero_at(10^-10)
        [ 1.0     1.0     1.0     1.0     1.0]
        [ 1.0     5.0    31.0   121.0   341.0]
        [ 1.0    31.0   341.0  1555.0  4681.0]
        [ 1.0   121.0  1555.0  7381.0 22621.0]
        [ 1.0   341.0  4681.0 22621.0 69905.0]

    A complex matrix that is Hermitian and positive definite.

        sage: A = matrix(CDF, [[        23,  17*I + 3,  24*I + 25,     21*I],
        ....:                  [ -17*I + 3,        38, -69*I + 89, 7*I + 15],
        ....:                  [-24*I + 25, 69*I + 89,        976, 24*I + 6],
        ....:                  [     -21*I, -7*I + 15,  -24*I + 6,       28]])
        sage: A.is_hermitian()
        True
        sage: L = A.cholesky()
        sage: L.round(6).zero_at(10^-10)
        [               4.795832                     0.0                    0.0       0.0]
        [  0.625543 - 3.544745*I                5.004346                    0.0       0.0]
        [   5.21286 - 5.004346*I 13.588189 + 10.721116*I              24.984023       0.0]
        [            -4.378803*I  -0.104257 - 0.851434*I  -0.21486 + 0.371348*I  2.811799]
        sage: (L*L.conjugate_transpose()).round(6).zero_at(10^-10)
        [         23.0  3.0 + 17.0*I 25.0 + 24.0*I        21.0*I]
        [ 3.0 - 17.0*I          38.0 89.0 - 69.0*I  15.0 + 7.0*I]
        [25.0 - 24.0*I 89.0 + 69.0*I         976.0  6.0 + 24.0*I]
        [      -21.0*I  15.0 - 7.0*I  6.0 - 24.0*I          28.0]

    This routine will recognize when the input matrix is not positive definite. The negative eigenvalues are an equivalent indicator. (Eigenvalues of a Hermitian matrix must be real, so there is no loss in ignoring the imprecise imaginary parts).

        sage: A = matrix(RDF, [[ 3,  -6,   9,   6,  -9],
        ....:                  [-6,  11, -16, -11,  17],
        ....:                  [ 9, -16,  28,  16, -40],
        ....:                  [ 6, -11,  16,   9, -19],
        ....:                  [-9,  17, -40, -19,  68]])
        sage: A.is_symmetric()
        True
        sage: A.eigenvalues()
        [108.07..., 13.02..., -0.02..., -0.70..., -1.37...]
        sage: A.cholesky()
        Traceback (most recent call last):
        ...
        ValueError: matrix is not positive definite

        sage: B = matrix(CDF, [[      2, 4 - 2*I, 2 + 2*I],
        ....:                  [4 + 2*I,       8,    10*I],
        ....:                  [2 - 2*I,   -10*I,      -3]])
        sage: B.is_hermitian()
        True
        sage: [ev.real() for ev in B.eigenvalues()]
        [15.88..., 0.08..., -8.97...]
        sage: B.cholesky()
        Traceback (most recent call last):
        ...
        ValueError: matrix is not positive definite

    TESTS:

    A trivial case.

        sage: A = matrix(RDF, 0, [])
        sage: A.cholesky()
        []

    The Cholesky factorization is only defined for square matrices.

        sage: A = matrix(RDF, 4, 5, range(20))
        sage: A.cholesky()
        Traceback (most recent call last):
        ...
        ValueError: Cholesky decomposition requires a square matrix, not a 4 x 5 matrix

  `condition`(p='frob')

    Returns the condition number of a square nonsingular matrix.

    Roughly speaking, this is a measure of how sensitive the matrix is to round-off errors in numerical computations. The minimum possible value is 1.0, and larger numbers indicate greater sensitivity.

    INPUT:

    - `p` - default: ‘frob’ - controls which norm is used to compute the condition number, allowable values are ‘frob’ (for the Frobenius norm), integers -2, -1, 1, 2, positive and negative infinity. See output discussion for specifics.

    OUTPUT:

    The condition number of a matrix is the product of a norm of the matrix times the norm of the inverse of the matrix. This requires that the matrix be square and invertible (nonsingular, full rank).

    Returned value is a double precision floating point value in `RDF`, or `Infinity`. Row and column sums described below are sums of the absolute values of the entries, where the absolute value of the complex number \(a+bi\) is \(\sqrt{a^2+b^2}\). Singular values are the “diagonal” entries of the “S” matrix in the singular value decomposition.

    - `p = 'frob'`: the default norm employed in computing the condition number, the Frobenius norm, which for a matrix \(A=(a_{ij})\) computes

      \[\left(\sum_{i,j}\left\lvert{a_{i,j}}\right\rvert^2\right)^{1/2}\]

    - `p = 'sv'`: the quotient of the maximal and minimal singular value.

    - `p = Infinity` or `p = oo`: the maximum row sum.

    - `p = -Infinity` or `p = -oo`: the minimum column sum.

    - `p = 1`: the maximum column sum.

    - `p = -1`: the minimum column sum.

    - `p = 2`: the 2-norm, equal to the maximum singular value.

    - `p = -2`: the minimum singular value.

    ALGORITHM:

    Computation is performed by the `cond()` function of the SciPy/NumPy library.

    EXAMPLES:

    First over the reals.

        sage: A = matrix(RDF, 4, [(1/4)*x^3 for x in range(16)]); A
        [   0.0   0.25    2.0   6.75]
        [  16.0  31.25   54.0  85.75]
        [ 128.0 182.25  250.0 332.75]
        [ 432.0 549.25  686.0 843.75]
        sage: A.condition()
        9923.88955...
        sage: A.condition(p='frob')
        9923.88955...
        sage: A.condition(p=Infinity)  # tol 2e-14
        22738.50000000045
        sage: A.condition(p=-Infinity)  # tol 2e-14
        17.50000000000028
        sage: A.condition(p=1)
        12139.21...
        sage: A.condition(p=-1)  # tol 2e-14
        550.0000000000093
        sage: A.condition(p=2)
        9897.8088...
        sage: A.condition(p=-2)
        0.000101032462...

    And over the complex numbers.

        sage: B = matrix(CDF, 3, [x + x^2*I for x in range(9)]); B
        [         0.0  1.0 + 1.0*I  2.0 + 4.0*I]
        [ 3.0 + 9.0*I 4.0 + 16.0*I 5.0 + 25.0*I]
        [6.0 + 36.0*I 7.0 + 49.0*I 8.0 + 64.0*I]
        sage: B.condition()
        203.851798...
        sage: B.condition(p='frob')
        203.851798...
        sage: B.condition(p=Infinity)
        369.55630...
        sage: B.condition(p=-Infinity)
        5.46112969...
        sage: B.condition(p=1)
        289.251481...
        sage: B.condition(p=-1)
        20.4566639...
        sage: B.condition(p=2)
        202.653543...
        sage: B.condition(p=-2)
        0.00493453005...

    Hilbert matrices are famously ill-conditioned, while an identity matrix can hit the minimum with the right norm.

        sage: A = matrix(RDF, 10, [1/(i+j+1) for i in range(10) for j in range(10)])
        sage: A.condition()  # tol 3e-5
        16332197709146.014
        sage: id = identity_matrix(CDF, 10)
        sage: id.condition(p=1)
        1.0

    Return values are in \(RDF\).

        sage: A = matrix(CDF, 2, range(1,5))
        sage: A.condition() in RDF
        True

    Rectangular and singular matrices raise errors if p is not ‘sv’.

        sage: A = matrix(RDF, 2, 3, range(6))
        sage: A.condition()
        Traceback (most recent call last):
        ...
        TypeError: matrix must be square if p is not 'sv', not 2 x 3

        sage: A.condition('sv')
        7.34...

        sage: A = matrix(QQ, 5, range(25))
        sage: A.is_singular()
        True
        sage: B = A.change_ring(CDF)
        sage: B.condition()
        Traceback (most recent call last):
        ...
        LinAlgError: Singular matrix

    Improper values of `p` are caught.

        sage: A = matrix(CDF, 2, range(1,5))
        sage: A.condition(p='bogus')
        Traceback (most recent call last):
        ...
        ValueError: condition number 'p' must be +/- infinity, 'frob', 'sv' or an integer, not bogus
        sage: A.condition(p=632)
        Traceback (most recent call last):
        ...
        ValueError: condition number integer values of 'p' must be -2, -1, 1 or 2, not 632

    TESTS:

    Some condition numbers, first by the definition which also exercises `norm()`, then by this method.

        sage: A = matrix(CDF, [[1,2,4],[5,3,9],[7,8,6]])
        sage: c = A.norm(2)*A.inverse().norm(2)
        sage: d = A.condition(2)
        sage: abs(c-d) < 1.0e-12
        True
        sage: c = A.norm(1)*A.inverse().norm(1)
        sage: d = A.condition(1)
        sage: abs(c-d) < 1.0e-12
        True

  `determinant`()

    Return the determinant of self.

    ALGORITHM:

    Use numpy

    EXAMPLES:

        sage: m = matrix(RDF,2,range(4)); m.det()
        -2.0
        sage: m = matrix(RDF,0,[]); m.det()
        1.0
        sage: m = matrix(RDF, 2, range(6)); m.det()
        Traceback (most recent call last):
        ...
        ValueError: self must be a square matrix

  `eigenvalues`(algorithm='default', tol=None)

    Returns a list of eigenvalues.

    INPUT:

    - `self` - a square matrix
    - `algorithm` - default: `'default'`
      - `'default'` - applicable to any matrix with double-precision floating point entries. Uses the `eigvals()` method from SciPy.
      - `'symmetric'` - converts the matrix into a real matrix (i.e. with entries from `RDF`), then applies the algorithm for Hermitian matrices. This algorithm can be significantly faster than the `'default'` algorithm.
      - `'hermitian'` - uses the `eigh()` method from SciPy, which applies only to real symmetric or complex Hermitian matrices. Since Hermitian is defined as a matrix equaling its conjugate-transpose, for a matrix with real entries this property is equivalent to being symmetric. This algorithm can be significantly faster than the `'default'` algorithm.
    - `'tol'` - default: `None` - if set to a value other than `None` this is interpreted as a small real number used to aid in grouping eigenvalues that are numerically similar. See the output description for more information.

    Warning

    When using the `'symmetric'` or `'hermitian'` algorithms, no check is made on the input matrix, and only the entries below, and on, the main diagonal are employed in the computation.

    Methods such as `is_symmetric()` and `is_hermitian()` could be used to verify this beforehand.

    OUTPUT:

    Default output for a square matrix of size \(n\) is a list of \(n\) eigenvalues from the complex double field, `CDF`. If the `'symmetric'` or `'hermitian'` algorithms are chosen, the returned eigenvalues are from the real double field, `RDF`.

    If a tolerance is specified, an attempt is made to group eigenvalues that are numerically similar. The return is then a list of pairs, where each pair is an eigenvalue followed by its multiplicity. The eigenvalue reported is the mean of the eigenvalues computed, and these eigenvalues are contained in an interval (or disk) whose radius is less than `5*tol` for \(n < 10,000\) in the worst case.

    More precisely, for an \(n\times n\) matrix, the diameter of the interval containing similar eigenvalues could be as large as sum of the reciprocals of the first \(n\) integers times `tol`.

    Warning

    Use caution when using the `tol` parameter to group eigenvalues. See the examples below to see how this can go wrong.

    EXAMPLES:

        sage: m = matrix(RDF, 2, 2, [1,2,3,4])
        sage: ev = m.eigenvalues(); ev
        [-0.372281323..., 5.37228132...]
        sage: ev[0].parent()
        Complex Double Field

        sage: m = matrix(RDF, 2, 2, [0,1,-1,0])
        sage: m.eigenvalues(algorithm='default')
        [1.0*I, -1.0*I]

        sage: m = matrix(CDF, 2, 2, [I,1,-I,0])
        sage: m.eigenvalues()
        [-0.624810533... + 1.30024259...*I, 0.624810533... - 0.30024259...*I]

    The adjacency matrix of a graph will be symmetric, and the eigenvalues will be real.

        sage: A = graphs.PetersenGraph().adjacency_matrix()
        sage: A = A.change_ring(RDF)
        sage: ev = A.eigenvalues(algorithm='symmetric'); ev  # tol 1e-14
        [-2.0000000000000004, -1.9999999999999998, -1.9999999999999998, -1.9999999999999993, 0.9999999999999994, 0.9999999999999997, 1.0, 1.0000000000000002, 1.0000000000000004, 2.9999999999999996]
        sage: ev[0].parent()
        Real Double Field

    The matrix `A` is “random”, but the construction of `B` provides a positive-definite Hermitian matrix. Note that the eigenvalues of a Hermitian matrix are real, and the eigenvalues of a positive-definite matrix will be positive.

        sage: A = matrix([[ 4*I + 5,  8*I + 1,  7*I + 5, 3*I + 5],
        ....:             [ 7*I - 2, -4*I + 7, -2*I + 4, 8*I + 8],
        ....:             [-2*I + 1,  6*I + 6,  5*I + 5,  -I - 4],
        ....:             [ 5*I + 1,  6*I + 2,    I - 4, -I + 3]])
        sage: B = (A*A.conjugate_transpose()).change_ring(CDF)
        sage: ev = B.eigenvalues(algorithm='hermitian'); ev
        [2.68144025..., 49.5167998..., 274.086188..., 390.71557...]
        sage: ev[0].parent()
        Real Double Field

    A tolerance can be given to aid in grouping eigenvalues that are similar numerically. However, if the parameter is too small it might split too finely. Too large, and it can go wrong very badly. Use with care.

        sage: G = graphs.PetersenGraph()
        sage: G.spectrum()
        [3, 1, 1, 1, 1, 1, -2, -2, -2, -2]

        sage: A = G.adjacency_matrix().change_ring(RDF)
        sage: A.eigenvalues(algorithm='symmetric', tol=1.0e-5)  # tol 1e-15
        [(-1.9999999999999998, 4), (1.0, 5), (2.9999999999999996, 1)]

        sage: A.eigenvalues(algorithm='symmetric', tol=2.5)  # tol 1e-15
        [(-1.9999999999999998, 4), (1.3333333333333333, 6)]

    An (extreme) example of properly grouping similar eigenvalues.

        sage: G = graphs.HigmanSimsGraph()
        sage: A = G.adjacency_matrix().change_ring(RDF)
        sage: A.eigenvalues(algorithm='symmetric', tol=1.0e-5)  # tol 2e-15
        [(-8.0, 22), (1.9999999999999984, 77), (21.999999999999996, 1)]

    TESTS:

    Testing bad input.

        sage: A = matrix(CDF, 2, range(4))
        sage: A.eigenvalues(algorithm='junk')
        Traceback (most recent call last):
        ...
        ValueError: algorithm must be 'default', 'symmetric', or 'hermitian', not junk

        sage: A = matrix(CDF, 2, 3, range(6))
        sage: A.eigenvalues()
        Traceback (most recent call last):
        ...
        ValueError: matrix must be square, not 2 x 3

        sage: A = matrix(CDF, 2, [1, 2, 3, 4*I])
        sage: A.eigenvalues(algorithm='symmetric')
        Traceback (most recent call last):
        ...
        TypeError: cannot apply symmetric algorithm to matrix with complex entries

        sage: A = matrix(CDF, 2, 2, range(4))
        sage: A.eigenvalues(tol='junk')
        Traceback (most recent call last):
        ...
        TypeError: tolerance parameter must be a real number, not junk

        sage: A = matrix(CDF, 2, 2, range(4))
        sage: A.eigenvalues(tol=-0.01)
        Traceback (most recent call last):
        ...
        ValueError: tolerance parameter must be positive, not -0.01

    A very small matrix.

        sage: matrix(CDF,0,0).eigenvalues()
        []

  `eigenvectors_left`()

    Compute the left eigenvectors of a matrix of double precision real or complex numbers (i.e. RDF or CDF).

    OUTPUT: Returns a list of triples, each of the form `(e,[v],1)`, where `e` is the eigenvalue, and `v` is an associated left eigenvector. If the matrix is of size \(n\), then there are \(n\) triples. Values are computed with the SciPy library.

    The format of this output is designed to match the format for exact results. However, since matrices here have numerical entries, the resulting eigenvalues will also be numerical. No attempt is made to determine if two eigenvalues are equal, or if eigenvalues might actually be zero. So the algebraic multiplicity of each eigenvalue is reported as 1. Decisions about equal eigenvalues or zero eigenvalues should be addressed in the calling routine.

    The SciPy routines used for these computations produce eigenvectors normalized to have length 1, but on different hardware they may vary by a sign. So for doctests we have normalized output by forcing their eigenvectors to have their first non-zero entry equal to one.

    EXAMPLES:

        sage: m = matrix(RDF, [[-5, 3, 2, 8],[10, 2, 4, -2],[-1, -10, -10, -17],[-2, 7, 6, 13]])
        sage: m
        [ -5.0   3.0   2.0   8.0]
        [ 10.0   2.0   4.0  -2.0]
        [ -1.0 -10.0 -10.0 -17.0]
        [ -2.0   7.0   6.0  13.0]
        sage: spectrum = m.left_eigenvectors()
        sage: for i in range(len(spectrum)):
        ....:   spectrum[i][1][0]=matrix(RDF, spectrum[i][1]).echelon_form()[0]
        sage: spectrum[0]  # tol 1e-13
        (2.0000000000000675, [(1.0, 1.0000000000000138, 1.0000000000000147, 1.0000000000000309)], 1)
        sage: spectrum[1]  # tol 1e-13
        (0.9999999999999164, [(0.9999999999999999, 0.7999999999999833, 0.7999999999999836, 0.5999999999999696)], 1)
        sage: spectrum[2]  # tol 1e-13
        (-1.9999999999999782, [(1.0, 0.40000000000000335, 0.6000000000000039, 0.2000000000000051)], 1)
        sage: spectrum[3]  # tol 1e-13
        (-1.0000000000000018, [(1.0, 0.9999999999999568, 1.9999999999998794, 1.9999999999998472)], 1)

  `eigenvectors_right`()

    Compute the right eigenvectors of a matrix of double precision real or complex numbers (i.e. RDF or CDF).

    OUTPUT:

    Returns a list of triples, each of the form `(e,[v],1)`, where `e` is the eigenvalue, and `v` is an associated right eigenvector. If the matrix is of size \(n\), then there are \(n\) triples. Values are computed with the SciPy library.

    The format of this output is designed to match the format for exact results. However, since matrices here have numerical entries, the resulting eigenvalues will also be numerical. No attempt is made to determine if two eigenvalues are equal, or if eigenvalues might actually be zero. So the algebraic multiplicity of each eigenvalue is reported as 1. Decisions about equal eigenvalues or zero eigenvalues should be addressed in the calling routine.

    The SciPy routines used for these computations produce eigenvectors normalized to have length 1, but on different hardware they may vary by a sign. So for doctests we have normalized output by forcing their eigenvectors to have their first non-zero entry equal to one.

    EXAMPLES:

        sage: m = matrix(RDF, [[-9, -14, 19, -74],[-1, 2, 4, -11],[-4, -12, 6, -32],[0, -2, -1, 1]])
        sage: m
        [ -9.0 -14.0  19.0 -74.0]
        [ -1.0   2.0   4.0 -11.0]
        [ -4.0 -12.0   6.0 -32.0]
        [  0.0  -2.0  -1.0   1.0]
        sage: spectrum = m.right_eigenvectors()
        sage: for i in range(len(spectrum)):
        ....:   spectrum[i][1][0]=matrix(RDF, spectrum[i][1]).echelon_form()[0]
        sage: spectrum[0]  # tol 1e-13
        (2.000000000000048, [(1.0, -2.0000000000001523, 3.000000000000181, 1.0000000000000746)], 1)
        sage: spectrum[1]  # tol 1e-13
        (0.999999999999941, [(1.0, -0.666666666666633, 1.333333333333286, 0.33333333333331555)], 1)
        sage: spectrum[2]  # tol 1e-13
        (-1.9999999999999483, [(1.0, -0.2000000000000063, 1.0000000000000173, 0.20000000000000498)], 1)
        sage: spectrum[3]  # tol 1e-13
        (-1.0000000000000406, [(1.0, -0.49999999999996264, 1.9999999999998617, 0.499999999999958)], 1)

  `exp`(algorithm=None, order=None)

    Calculate the exponential of this matrix X, which is the matrix

    \[e^X = \sum_{k=0}^{\infty} \frac{X^k}{k!}.\]

    INPUT:

    - algorithm – deprecated
    - order – deprecated

    EXAMPLES:

        sage: A=matrix(RDF, 2, [1,2,3,4]); A
        [1.0 2.0]
        [3.0 4.0]
        sage: A.exp()  # tol 1e-15
        [51.968956198705044  74.73656456700327]
        [112.10484685050491 164.07380304920997]
        sage: A=matrix(CDF, 2, [1,2+I,3*I,4]); A
        [        1.0 2.0 + 1.0*I]
        [      3.0*I         4.0]
        sage: A.exp()  # tol 1e-14
        [-19.614602953804912 + 12.517743846762578*I   3.7949636449582176 + 28.88379930658099*I]
        [ -32.383580980922254 + 21.88423595789845*I   2.269633004093535 + 44.901324827684824*I]

    TESTS:

        sage: A = matrix(RDF, 2, [1,2,3,4])
        sage: E = A.exp(algorithm='eig')
        doctest:...: DeprecationWarning: The algorithm and order arguments are deprecated.
        See http://trac.sagemath.org/17140 for details.
        sage: E  # tol 1e-15
        [51.968956198705044  74.73656456700327]
        [112.10484685050491 164.07380304920997]
        sage: A.exp(algorithm='taylor')   # tol 1e-15
        [51.968956198705044  74.73656456700327]
        [112.10484685050491 164.07380304920997]
        sage: A = matrix(CDF, 2, [1,2+I,3*I,4])
        sage: A.exp(algorithm='eig')  # tol 3e-14
        [-19.614602953804923 + 12.51774384676257*I 3.7949636449582016 + 28.883799306580997*I]
        [-32.38358098092227 + 21.884235957898433*I  2.2696330040935084 + 44.90132482768484*I]

  `is_hermitian`(tol=1e-12, algorithm='orthonormal')

    Returns `True` if the matrix is equal to its conjugate-transpose.

    INPUT:

    - `tol` - default: `1e-12` - the largest value of the absolute value of the difference between two matrix entries for which they will still be considered equal.
    - `algorithm` - default: ‘orthonormal’ - set to ‘orthonormal’ for a stable procedure and set to ‘naive’ for a fast procedure.

    OUTPUT:

    `True` if the matrix is square and equal to the transpose with every entry conjugated, and `False` otherwise.

    Note that if conjugation has no effect on elements of the base ring (such as for integers), then the `is_symmetric()` method is equivalent and faster.

    The tolerance parameter is used to allow for numerical values to be equal if there is a slight difference due to round-off and other imprecisions.

    The result is cached, on a per-tolerance and per-algorithm basis.

    ALGORITHMS:

    The naive algorithm simply compares corresponing entries on either side of the diagonal (and on the diagonal itself) to see if they are conjugates, with equality controlled by the tolerance parameter.

    The orthonormal algorithm first computes a Schur decomposition (via the `schur()` method) and checks that the result is a diagonal matrix with real entries.

    So the naive algorithm can finish quickly for a matrix that is not Hermitian, while the orthonormal algorithm will always compute a Schur decomposition before going through a similar check of the matrix entry-by-entry.

    EXAMPLES:

        sage: A = matrix(CDF, [[ 1 + I,  1 - 6*I, -1 - I],
        ....:                  [-3 - I,     -4*I,     -2],
        ....:                  [-1 + I, -2 - 8*I,  2 + I]])
        sage: A.is_hermitian(algorithm='orthonormal')
        False
        sage: A.is_hermitian(algorithm='naive')
        False
        sage: B = A*A.conjugate_transpose()
        sage: B.is_hermitian(algorithm='orthonormal')
        True
        sage: B.is_hermitian(algorithm='naive')
        True

    A matrix that is nearly Hermitian, but for one non-real diagonal entry.

        sage: A = matrix(CDF, [[    2,   2-I, 1+4*I],
        ....:                  [  2+I,   3+I, 2-6*I],
        ....:                  [1-4*I, 2+6*I,     5]])
        sage: A.is_hermitian(algorithm='orthonormal')
        False
        sage: A[1,1] = 132
        sage: A.is_hermitian(algorithm='orthonormal')
        True

    We get a unitary matrix from the SVD routine and use this numerical matrix to create a matrix that should be Hermitian (indeed it should be the identity matrix), but with some imprecision. We use this to illustrate that if the tolerance is set too small, then we can be too strict about the equality of entries and may achieve the wrong result (depending on the system):

        sage: A = matrix(CDF, [[ 1 + I,  1 - 6*I, -1 - I],
        ....:                  [-3 - I,     -4*I,     -2],
        ....:                  [-1 + I, -2 - 8*I,  2 + I]])
        sage: U, _, _ = A.SVD()
        sage: B=U*U.conjugate_transpose()
        sage: B.is_hermitian(algorithm='naive')
        True
        sage: B.is_hermitian(algorithm='naive', tol=1.0e-17)  # random
        False
        sage: B.is_hermitian(algorithm='naive', tol=1.0e-15)
        True

    A square, empty matrix is trivially Hermitian.

        sage: A = matrix(RDF, 0, 0)
        sage: A.is_hermitian()
        True

    Rectangular matrices are never Hermitian, no matter which algorithm is requested.

        sage: A = matrix(CDF, 3, 4)
        sage: A.is_hermitian()
        False

    TESTS:

    The tolerance must be strictly positive.

        sage: A = matrix(RDF, 2, range(4))
        sage: A.is_hermitian(tol = -3.1)
        Traceback (most recent call last):
        ...
        ValueError: tolerance must be positive, not -3.1

    The `algorithm` keyword gets checked.

        sage: A = matrix(RDF, 2, range(4))
        sage: A.is_hermitian(algorithm='junk')
        Traceback (most recent call last):
        ...
        ValueError: algorithm must be 'naive' or 'orthonormal', not junk

    AUTHOR:

    - Rob Beezer (2011-03-30)
  `is_normal`(tol=1e-12, algorithm='orthonormal')

    Returns `True` if the matrix commutes with its conjugate-transpose.

    INPUT:

    - `tol` - default: `1e-12` - the largest value of the absolute value of the difference between two matrix entries for which they will still be considered equal.
    - `algorithm` - default: ‘orthonormal’ - set to ‘orthonormal’ for a stable procedure and set to ‘naive’ for a fast procedure.

    OUTPUT:

    `True` if the matrix is square and commutes with its conjugate-transpose, and `False` otherwise.

    Normal matrices are precisely those that can be diagonalized by a unitary matrix.

    The tolerance parameter is used to allow for numerical values to be equal if there is a slight difference due to round-off and other imprecisions.

    The result is cached, on a per-tolerance and per-algorithm basis.

    ALGORITHMS:

    The naive algorithm simply compares entries of the two possible products of the matrix with its conjugate-transpose, with equality controlled by the tolerance parameter.

    The orthonormal algorithm first computes a Schur decomposition (via the `schur()` method) and checks that the result is a diagonal matrix. An orthonormal diagonalization is equivalent to being normal.

    So the naive algorithm can finish fairly quickly for a matrix that is not normal, once the products have been computed. However, the orthonormal algorithm will compute a Schur decomposition before going through a similar check of a matrix entry-by-entry.

    EXAMPLES:

    First over the complexes. `B` is Hermitian, hence normal.

        sage: A = matrix(CDF, [[ 1 + I,  1 - 6*I, -1 - I],
        ....:                  [-3 - I,     -4*I,     -2],
        ....:                  [-1 + I, -2 - 8*I,  2 + I]])
        sage: B = A*A.conjugate_transpose()
        sage: B.is_hermitian()
        True
        sage: B.is_normal(algorithm='orthonormal')
        True
        sage: B.is_normal(algorithm='naive')
        True
        sage: B[0,0] = I
        sage: B.is_normal(algorithm='orthonormal')
        False
        sage: B.is_normal(algorithm='naive')
        False

    Now over the reals. Circulant matrices are normal.

        sage: G = graphs.CirculantGraph(20, [3, 7])
        sage: D = digraphs.Circuit(20)
        sage: A = 3*D.adjacency_matrix() - 5*G.adjacency_matrix()
        sage: A = A.change_ring(RDF)
        sage: A.is_normal()
        True
        sage: A.is_normal(algorithm = 'naive')
        True
        sage: A[19,0] = 4.0
        sage: A.is_normal()
        False
        sage: A.is_normal(algorithm = 'naive')
        False

    Skew-Hermitian matrices are normal.

        sage: A = matrix(CDF, [[ 1 + I,  1 - 6*I, -1 - I],
        ....:                  [-3 - I,     -4*I,     -2],
        ....:                  [-1 + I, -2 - 8*I,  2 + I]])
        sage: B = A - A.conjugate_transpose()
        sage: B.is_hermitian()
        False
        sage: B.is_normal()
        True
        sage: B.is_normal(algorithm='naive')
        True

    A small matrix that does not fit into any of the usual categories of normal matrices.

        sage: A = matrix(RDF, [[1, -1],
        ....:                  [1,  1]])
        sage: A.is_normal()
        True
        sage: not A.is_hermitian() and not A.is_skew_symmetric()
        True

    Sage has several fields besides the entire complex numbers where conjugation is non-trivial.

        sage: F.<b> = QuadraticField(-7)
        sage: C = matrix(F, [[-2*b - 3,  7*b - 6, -b + 3],
        ....:                [-2*b - 3, -3*b + 2,   -2*b],
        ....:                [   b + 1,        0,     -2]])
        sage: C = C*C.conjugate_transpose()
        sage: C.is_normal()
        True

    A square, empty matrix is trivially normal.

        sage: A = matrix(CDF, 0, 0)
        sage: A.is_normal()
        True

    Rectangular matrices are never normal, no matter which algorithm is requested.

        sage: A = matrix(CDF, 3, 4)
        sage: A.is_normal()
        False

    TESTS:

    The tolerance must be strictly positive.

        sage: A = matrix(RDF, 2, range(4))
        sage: A.is_normal(tol = -3.1)
        Traceback (most recent call last):
        ...
        ValueError: tolerance must be positive, not -3.1

    The `algorithm` keyword gets checked.

        sage: A = matrix(RDF, 2, range(4))
        sage: A.is_normal(algorithm='junk')
        Traceback (most recent call last):
        ...
        ValueError: algorithm must be 'naive' or 'orthonormal', not junk

    AUTHOR:

    > - Rob Beezer (2011-03-31)

  `is_positive_definite`()

    Determines if a matrix is positive definite.

    A matrix \(A\) is positive definite if it is square, is Hermitian (which reduces to symmetric in the real case), and for every nonzero vector \(\vec{x}\),

    \[\begin{split}\vec{x}^\ast A \vec{x} > 0\end{split}\]

    where \(\vec{x}^\ast\) is the conjugate-transpose in the complex case and just the transpose in the real case. Equivalently, a positive definite matrix has only positive eigenvalues and only positive determinants of leading principal submatrices.

    INPUT:

    Any matrix over `RDF` or `CDF`.

    OUTPUT:

    `True` if and only if the matrix is square, Hermitian, and meets the condition above on the quadratic form. The result is cached.

    IMPLEMENTATION:

    The existence of a Cholesky decomposition and the positive definite property are equivalent. So this method and the `cholesky()` method compute and cache both the Cholesky decomposition and the positive-definiteness. So the `is_positive_definite()` method or catching a `ValueError` from the `cholesky()` method are equally expensive computationally and if the decomposition exists, it is cached as a side-effect of either routine.

    EXAMPLES:

    A matrix over `RDF` that is positive definite.

        sage: M = matrix(RDF,[[ 1,  1,    1,     1,     1],
        ....:                 [ 1,  5,   31,   121,   341],
        ....:                 [ 1, 31,  341,  1555,  4681],
        ....:                 [ 1,121, 1555,  7381, 22621],
        ....:                 [ 1,341, 4681, 22621, 69905]])
        sage: M.is_symmetric()
        True
        sage: M.eigenvalues()
        [77547.66..., 82.44..., 2.41..., 0.46..., 0.011...]
        sage: [round(M[:i,:i].determinant()) for i in range(1, M.nrows()+1)]
        [1, 4, 460, 27936, 82944]
        sage: M.is_positive_definite()
        True

    A matrix over `CDF` that is positive definite.

        sage: C = matrix(CDF, [[        23,  17*I + 3,  24*I + 25,     21*I],
        ....:                  [ -17*I + 3,        38, -69*I + 89, 7*I + 15],
        ....:                  [-24*I + 25, 69*I + 89,        976, 24*I + 6],
        ....:                  [     -21*I, -7*I + 15,  -24*I + 6,       28]])
        sage: C.is_hermitian()
        True
        sage: [x.real() for x in C.eigenvalues()]
        [991.46..., 55.96..., 3.69..., 13.87...]
        sage: [round(C[:i,:i].determinant().real()) for i in range(1, C.nrows()+1)]
        [23, 576, 359540, 2842600]
        sage: C.is_positive_definite()
        True

    A matrix over `RDF` that is not positive definite.

        sage: A = matrix(RDF, [[ 3,  -6,   9,   6,  -9],
        ....:                  [-6,  11, -16, -11,  17],
        ....:                  [ 9, -16,  28,  16, -40],
        ....:                  [ 6, -11,  16,   9, -19],
        ....:                  [-9,  17, -40, -19,  68]])
        sage: A.is_symmetric()
        True
        sage: A.eigenvalues()
        [108.07..., 13.02..., -0.02..., -0.70..., -1.37...]
        sage: [round(A[:i,:i].determinant()) for i in range(1, A.nrows()+1)]
        [3, -3, -15, 30, -30]
        sage: A.is_positive_definite()
        False

    A matrix over `CDF` that is not positive definite.

        sage: B = matrix(CDF, [[      2, 4 - 2*I, 2 + 2*I],
        ....:                  [4 + 2*I,       8,    10*I],
        ....:                  [2 - 2*I,   -10*I,      -3]])
        sage: B.is_hermitian()
        True
        sage: [ev.real() for ev in B.eigenvalues()]
        [15.88..., 0.08..., -8.97...]
        sage: [round(B[:i,:i].determinant().real()) for i in range(1, B.nrows()+1)]
        [2, -4, -12]
        sage: B.is_positive_definite()
        False

    A large random matrix that is guaranteed by theory to be positive definite.

        sage: R = random_matrix(CDF, 200)
        sage: H = R.conjugate_transpose()*R
        sage: H.is_positive_definite()
        True

    TESTS:

    A trivially small case.

        sage: S = matrix(CDF, [])
        sage: S.nrows(), S.ncols()
        (0, 0)
        sage: S.is_positive_definite()
        True

    A rectangular matrix will never be positive definite.

        sage: R = matrix(RDF, 2, 3, range(6))
        sage: R.is_positive_definite()
        False

    A non-Hermitian matrix will never be positive definite.

        sage: T = matrix(CDF, 8, 8, range(64))
        sage: T.is_positive_definite()
        False

    AUTHOR:

    - Rob Beezer (2012-05-28)
  `is_symmetric`(tol=1e-12)

    Return whether this matrix is symmetric, to the given tolerance.

    EXAMPLES:

        sage: m = matrix(RDF,2,2,range(4)); m
        [0.0 1.0]
        [2.0 3.0]
        sage: m.is_symmetric()
        False
        sage: m[1,0] = 1.1; m
        [0.0 1.0]
        [1.1 3.0]
        sage: m.is_symmetric()
        False

    The tolerance inequality is strict:
      sage: m.is_symmetric(tol=0.1) False sage: m.is_symmetric(tol=0.11) True
  `is_unitary`(tol=1e-12, algorithm='orthonormal')

    Returns `True` if the columns of the matrix are an orthonormal basis.

    For a matrix with real entries this determines if a matrix is “orthogonal” and for a matrix with complex entries this determines if the matrix is “unitary.”

    INPUT:

    - `tol` - default: `1e-12` - the largest value of the absolute value of the difference between two matrix entries for which they will still be considered equal.
    - `algorithm` - default: ‘orthonormal’ - set to ‘orthonormal’ for a stable procedure and set to ‘naive’ for a fast procedure.

    OUTPUT:

    `True` if the matrix is square and its conjugate-transpose is its inverse, and `False` otherwise. In other words, a matrix is orthogonal or unitary if the product of its conjugate-transpose times the matrix is the identity matrix.

    The tolerance parameter is used to allow for numerical values to be equal if there is a slight difference due to round-off and other imprecisions.

    The result is cached, on a per-tolerance and per-algorithm basis.

    ALGORITHMS:

    The naive algorithm simply computes the product of the conjugate-transpose with the matrix and compares the entries to the identity matrix, with equality controlled by the tolerance parameter.

    The orthonormal algorithm first computes a Schur decomposition (via the `schur()` method) and checks that the result is a diagonal matrix with entries of modulus 1, which is equivalent to being unitary.

    So the naive algorithm might finish fairly quickly for a matrix that is not unitary, once the product has been computed. However, the orthonormal algorithm will compute a Schur decomposition before going through a similar check of a matrix entry-by-entry.

    EXAMPLES:

    A matrix that is far from unitary.

        sage: A = matrix(RDF, 4, range(16))
        sage: A.conjugate().transpose()*A
        [224.0 248.0 272.0 296.0]
        [248.0 276.0 304.0 332.0]
        [272.0 304.0 336.0 368.0]
        [296.0 332.0 368.0 404.0]
        sage: A.is_unitary()
        False
        sage: A.is_unitary(algorithm='naive')
        False
        sage: A.is_unitary(algorithm='orthonormal')
        False

    The QR decoposition will produce a unitary matrix as Q and the SVD decomposition will create two unitary matrices, U and V.

        sage: A = matrix(CDF, [[   1 - I,   -3*I,  -2 + I,        1, -2 + 3*I],
        ....:                  [   1 - I, -2 + I, 1 + 4*I,        0,    2 + I],
        ....:                  [      -1, -5 + I,  -2 + I,    1 + I, -5 - 4*I],
        ....:                  [-2 + 4*I,  2 - I, 8 - 4*I,  1 - 8*I,  3 - 2*I]])
        sage: Q, R = A.QR()
        sage: Q.is_unitary()
        True
        sage: U, S, V = A.SVD()
        sage: U.is_unitary(algorithm='naive')
        True
        sage: U.is_unitary(algorithm='orthonormal')
        True
        sage: V.is_unitary(algorithm='naive')  # not tested - known bug (trac #11248)
        True

    If we make the tolerance too strict we can get misleading results.

        sage: A = matrix(RDF, 10, 10, [1/(i+j+1) for i in range(10) for j in range(10)])
        sage: Q, R = A.QR()
        sage: Q.is_unitary(algorithm='naive', tol=1e-16)
        False
        sage: Q.is_unitary(algorithm='orthonormal', tol=1e-17)
        False

    Rectangular matrices are not unitary/orthogonal, even if their columns form an orthonormal set.

        sage: A = matrix(CDF, [[1,0], [0,0], [0,1]])
        sage: A.is_unitary()
        False

    The smallest cases. The Schur decomposition used by the orthonormal algorithm will fail on a matrix of size zero.

        sage: P = matrix(CDF, 0, 0)
        sage: P.is_unitary(algorithm='naive')
        True

        sage: P = matrix(CDF, 1, 1, [1])
        sage: P.is_unitary(algorithm='orthonormal')
        True

        sage: P = matrix(CDF, 0, 0,)
        sage: P.is_unitary(algorithm='orthonormal')
        Traceback (most recent call last):
        ...
        ValueError: failed to create intent(cache|hide)|optional array-- must have defined dimensions but got (0,)

    TESTS:

        sage: P = matrix(CDF, 2, 2)
        sage: P.is_unitary(tol='junk')
        Traceback (most recent call last):
        ...
        TypeError: tolerance must be a real number, not junk

        sage: P.is_unitary(tol=-0.3)
        Traceback (most recent call last):
        ...
        ValueError: tolerance must be positive, not -0.3

        sage: P.is_unitary(algorithm='junk')
        Traceback (most recent call last):
        ...
        ValueError: algorithm must be 'naive' or 'orthonormal', not junk

    AUTHOR:

    - Rob Beezer (2011-05-04)
  `left_eigenvectors`()

    Compute the left eigenvectors of a matrix of double precision real or complex numbers (i.e. RDF or CDF).

    OUTPUT: Returns a list of triples, each of the form `(e,[v],1)`, where `e` is the eigenvalue, and `v` is an associated left eigenvector. If the matrix is of size \(n\), then there are \(n\) triples. Values are computed with the SciPy library.

    The format of this output is designed to match the format for exact results. However, since matrices here have numerical entries, the resulting eigenvalues will also be numerical. No attempt is made to determine if two eigenvalues are equal, or if eigenvalues might actually be zero. So the algebraic multiplicity of each eigenvalue is reported as 1. Decisions about equal eigenvalues or zero eigenvalues should be addressed in the calling routine.

    The SciPy routines used for these computations produce eigenvectors normalized to have length 1, but on different hardware they may vary by a sign. So for doctests we have normalized output by forcing their eigenvectors to have their first non-zero entry equal to one.

    EXAMPLES:

        sage: m = matrix(RDF, [[-5, 3, 2, 8],[10, 2, 4, -2],[-1, -10, -10, -17],[-2, 7, 6, 13]])
        sage: m
        [ -5.0   3.0   2.0   8.0]
        [ 10.0   2.0   4.0  -2.0]
        [ -1.0 -10.0 -10.0 -17.0]
        [ -2.0   7.0   6.0  13.0]
        sage: spectrum = m.left_eigenvectors()
        sage: for i in range(len(spectrum)):
        ....:   spectrum[i][1][0]=matrix(RDF, spectrum[i][1]).echelon_form()[0]
        sage: spectrum[0]  # tol 1e-13
        (2.0000000000000675, [(1.0, 1.0000000000000138, 1.0000000000000147, 1.0000000000000309)], 1)
        sage: spectrum[1]  # tol 1e-13
        (0.9999999999999164, [(0.9999999999999999, 0.7999999999999833, 0.7999999999999836, 0.5999999999999696)], 1)
        sage: spectrum[2]  # tol 1e-13
        (-1.9999999999999782, [(1.0, 0.40000000000000335, 0.6000000000000039, 0.2000000000000051)], 1)
        sage: spectrum[3]  # tol 1e-13
        (-1.0000000000000018, [(1.0, 0.9999999999999568, 1.9999999999998794, 1.9999999999998472)], 1)

  `log_determinant`()

    Compute the log of the absolute value of the determinant using LU decomposition.

    Note

    This is useful if the usual determinant overflows.

    EXAMPLES:

        sage: m = matrix(RDF,2,2,range(4)); m
        [0.0 1.0]
        [2.0 3.0]
        sage: RDF(log(abs(m.determinant())))
        0.6931471805599453
        sage: m.log_determinant()
        0.6931471805599453
        sage: m = matrix(RDF,0,0,[]); m
        []
        sage: m.log_determinant()
        0.0
        sage: m = matrix(CDF,2,2,range(4)); m
        [0.0 1.0]
        [2.0 3.0]
        sage: RDF(log(abs(m.determinant())))
        0.6931471805599453
        sage: m.log_determinant()
        0.6931471805599453
        sage: m = matrix(CDF,0,0,[]); m
        []
        sage: m.log_determinant()
        0.0

  `norm`(p=2)

    Returns the norm of the matrix.

    INPUT:

    - `p` - default: 2 - controls which norm is computed, allowable values are ‘frob’ (for the Frobenius norm), integers -2, -1, 1, 2, positive and negative infinity. See output discussion for specifics.

    OUTPUT:

    Returned value is a double precision floating point value in `RDF`. Row and column sums described below are sums of the absolute values of the entries, where the absolute value of the complex number \(a+bi\) is \(\sqrt{a^2+b^2}\). Singular values are the “diagonal” entries of the “S” matrix in the singular value decomposition.

    - `p = 'frob'`: the Frobenius norm, which for a matrix \(A=(a_{ij})\) computes

      \[\left(\sum_{i,j}\left\lvert{a_{i,j}}\right\rvert^2\right)^{1/2}\]

    - `p = Infinity` or `p = oo`: the maximum row sum.

    - `p = -Infinity` or `p = -oo`: the minimum column sum.

    - `p = 1`: the maximum column sum.

    - `p = -1`: the minimum column sum.

    - `p = 2`: the induced 2-norm, equal to the maximum singular value.

    - `p = -2`: the minimum singular value.

    ALGORITHM:

    Computation is performed by the `norm()` function of the SciPy/NumPy library.

    EXAMPLES:

    First over the reals.

        sage: A = matrix(RDF, 3, range(-3, 6)); A
        [-3.0 -2.0 -1.0]
        [ 0.0  1.0  2.0]
        [ 3.0  4.0  5.0]
        sage: A.norm()
        7.99575670...
        sage: A.norm(p='frob')
        8.30662386...
        sage: A.norm(p=Infinity)
        12.0
        sage: A.norm(p=-Infinity)
        3.0
        sage: A.norm(p=1)
        8.0
        sage: A.norm(p=-1)
        6.0
        sage: A.norm(p=2)
        7.99575670...
        sage: A.norm(p=-2) < 10^-15
        True

    And over the complex numbers.

        sage: B = matrix(CDF, 2, [[1+I, 2+3*I],[3+4*I,3*I]]); B
        [1.0 + 1.0*I 2.0 + 3.0*I]
        [3.0 + 4.0*I       3.0*I]
        sage: B.norm()
        6.66189877...
        sage: B.norm(p='frob')
        7.0
        sage: B.norm(p=Infinity)
        8.0
        sage: B.norm(p=-Infinity)
        5.01976483...
        sage: B.norm(p=1)
        6.60555127...
        sage: B.norm(p=-1)
        6.41421356...
        sage: B.norm(p=2)
        6.66189877...
        sage: B.norm(p=-2)
        2.14921023...

    Since it is invariant under unitary multiplication, the Frobenius norm is equal to the square root of the sum of squares of the singular values.

        sage: A = matrix(RDF, 5, range(1,26))
        sage: f = A.norm(p='frob')
        sage: U, S, V = A.SVD()
        sage: s = sqrt(sum([S[i,i]^2 for i in range(5)]))
        sage: abs(f-s) < 1.0e-12
        True

    Return values are in \(RDF\).

        sage: A = matrix(CDF, 2, range(4))
        sage: A.norm() in RDF
        True

    Improper values of `p` are caught.

        sage: A.norm(p='bogus')
        Traceback (most recent call last):
        ...
        ValueError: matrix norm 'p' must be +/- infinity, 'frob' or an integer, not bogus
        sage: A.norm(p=632)
        Traceback (most recent call last):
        ...
        ValueError: matrix norm integer values of 'p' must be -2, -1, 1 or 2, not 632

  `numpy`(dtype=None)

    This method returns a copy of the matrix as a numpy array. It uses the numpy C/api so is very fast.

    INPUT:

    - `dtype` - The desired data-type for the array. If not given, then the type will be determined as the minimum type required to hold the objects in the sequence.

    EXAMPLES:

        sage: m = matrix(RDF,[[1,2],[3,4]])
        sage: n = m.numpy()
        sage: import numpy
        sage: numpy.linalg.eig(n)
        (array([-0.37228132,  5.37228132]), array([[-0.82456484, -0.41597356],
               [ 0.56576746, -0.90937671]]))
        sage: m = matrix(RDF, 2, range(6)); m
        [0.0 1.0 2.0]
        [3.0 4.0 5.0]
        sage: m.numpy()
        array([[ 0.,  1.,  2.],
               [ 3.,  4.,  5.]])

    Alternatively, numpy automatically calls this function (via the magic `__array__()` method) to convert Sage matrices to numpy arrays:

        sage: import numpy
        sage: m = matrix(RDF, 2, range(6)); m
        [0.0 1.0 2.0]
        [3.0 4.0 5.0]
        sage: numpy.array(m)
        array([[ 0.,  1.,  2.],
               [ 3.,  4.,  5.]])
        sage: numpy.array(m).dtype
        dtype('float64')
        sage: m = matrix(CDF, 2, range(6)); m
        [0.0 1.0 2.0]
        [3.0 4.0 5.0]
        sage: numpy.array(m)
        array([[ 0.+0.j,  1.+0.j,  2.+0.j],
               [ 3.+0.j,  4.+0.j,  5.+0.j]])
        sage: numpy.array(m).dtype
        dtype('complex128')

    TESTS:

        sage: m = matrix(RDF,0,5,[]); m
        []
        sage: m.numpy()
        array([], shape=(0, 5), dtype=float64)
        sage: m = matrix(RDF,5,0,[]); m
        []
        sage: m.numpy()
        array([], shape=(5, 0), dtype=float64)

  `right_eigenvectors`()

    Compute the right eigenvectors of a matrix of double precision real or complex numbers (i.e. RDF or CDF).

    OUTPUT:

    Returns a list of triples, each of the form `(e,[v],1)`, where `e` is the eigenvalue, and `v` is an associated right eigenvector. If the matrix is of size \(n\), then there are \(n\) triples. Values are computed with the SciPy library.

    The format of this output is designed to match the format for exact results. However, since matrices here have numerical entries, the resulting eigenvalues will also be numerical. No attempt is made to determine if two eigenvalues are equal, or if eigenvalues might actually be zero. So the algebraic multiplicity of each eigenvalue is reported as 1. Decisions about equal eigenvalues or zero eigenvalues should be addressed in the calling routine.

    The SciPy routines used for these computations produce eigenvectors normalized to have length 1, but on different hardware they may vary by a sign. So for doctests we have normalized output by forcing their eigenvectors to have their first non-zero entry equal to one.

    EXAMPLES:

        sage: m = matrix(RDF, [[-9, -14, 19, -74],[-1, 2, 4, -11],[-4, -12, 6, -32],[0, -2, -1, 1]])
        sage: m
        [ -9.0 -14.0  19.0 -74.0]
        [ -1.0   2.0   4.0 -11.0]
        [ -4.0 -12.0   6.0 -32.0]
        [  0.0  -2.0  -1.0   1.0]
        sage: spectrum = m.right_eigenvectors()
        sage: for i in range(len(spectrum)):
        ....:   spectrum[i][1][0]=matrix(RDF, spectrum[i][1]).echelon_form()[0]
        sage: spectrum[0]  # tol 1e-13
        (2.000000000000048, [(1.0, -2.0000000000001523, 3.000000000000181, 1.0000000000000746)], 1)
        sage: spectrum[1]  # tol 1e-13
        (0.999999999999941, [(1.0, -0.666666666666633, 1.333333333333286, 0.33333333333331555)], 1)
        sage: spectrum[2]  # tol 1e-13
        (-1.9999999999999483, [(1.0, -0.2000000000000063, 1.0000000000000173, 0.20000000000000498)], 1)
        sage: spectrum[3]  # tol 1e-13
        (-1.0000000000000406, [(1.0, -0.49999999999996264, 1.9999999999998617, 0.499999999999958)], 1)

  `round`(ndigits=0)

    Returns a copy of the matrix where all entries have been rounded to a given precision in decimal digits (default 0 digits).

    INPUT:

    - `ndigits` - The precision in number of decimal digits

    OUTPUT:

    A modified copy of the matrix

    EXAMPLES:

        sage: M = matrix(CDF, [[10.234r + 34.2343jr, 34e10r]])
        sage: M
        [10.234 + 34.2343*I     340000000000.0]
        sage: M.round(2)
        [10.23 + 34.23*I  340000000000.0]
        sage: M.round()
        [ 10.0 + 34.0*I 340000000000.0]

  `schur`(base_ring=None)

    Returns the Schur decomposition of the matrix.

    INPUT:

    - `base_ring` - optional, defaults to the base ring of `self`. Use this to request the base ring of the returned matrices, which will affect the format of the results.

    OUTPUT:

    A pair of immutable matrices. The first is a unitary matrix \(Q\). The second, \(T\), is upper-triangular when returned over the complex numbers, while it is almost upper-triangular over the reals. In the latter case, there can be some \(2\times 2\) blocks on the diagonal which represent a pair of conjugate complex eigenvalues of `self`.

    If `self` is the matrix \(A\), then

    \[A = QT({\overline Q})^t\]

    where the latter matrix is the conjugate-transpose of `Q`, which is also the inverse of `Q`, since `Q` is unitary.

    Note that in the case of a normal matrix (Hermitian, symmetric, and others), the upper-triangular matrix is a diagonal matrix with eigenvalues of `self` on the diagonal, and the unitary matrix has columns that form an orthonormal basis composed of eigenvectors of `self`. This is known as “orthonormal diagonalization”.

    Warning

    The Schur decomposition is not unique, as there may be numerous choices for the vectors of the orthonormal basis, and consequently different possibilities for the upper-triangular matrix. However, the diagonal of the upper-triangular matrix will always contain the eigenvalues of the matrix (in the complex version), or \(2\times 2\) block matrices in the real version representing pairs of conjugate complex eigenvalues.

    In particular, results may vary across systems and processors.

    EXAMPLES:

    First over the complexes. The similar matrix is always upper-triangular in this case.

        sage: A = matrix(CDF, 4, 4, range(16)) + matrix(CDF, 4, 4, [x^3*I for x in range(0, 16)])
        sage: Q, T = A.schur()
        sage: (Q*Q.conjugate().transpose()).zero_at(1e-12)  # tol 1e-12
        [ 0.999999999999999                0.0                0.0                0.0]
        [               0.0 0.9999999999999996                0.0                0.0]
        [               0.0                0.0 0.9999999999999992                0.0]
        [               0.0                0.0                0.0 0.9999999999999999]
        sage: all([T.zero_at(1.0e-12)[i,j] == 0 for i in range(4) for j in range(i)])
        True
        sage: (Q*T*Q.conjugate().transpose()-A).zero_at(1.0e-11)
        [0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0]
        sage: eigenvalues = [T[i,i] for i in range(4)]; eigenvalues
        [30.733... + 4648.541...*I, -0.184... - 159.057...*I, -0.523... + 11.158...*I, -0.025... - 0.642...*I]
        sage: A.eigenvalues()
        [30.733... + 4648.541...*I, -0.184... - 159.057...*I, -0.523... + 11.158...*I, -0.025... - 0.642...*I]
        sage: abs(A.norm()-T.norm()) < 1e-10
        True

    We begin with a real matrix but ask for a decomposition over the complexes. The result will yield an upper-triangular matrix over the complex numbers for `T`.

        sage: A = matrix(RDF, 4, 4, [x^3 for x in range(16)])
        sage: Q, T = A.schur(base_ring=CDF)
        sage: (Q*Q.conjugate().transpose()).zero_at(1e-12)  # tol 1e-12
        [0.9999999999999987                0.0                0.0                0.0]
        [               0.0 0.9999999999999999                0.0                0.0]
        [               0.0                0.0 1.0000000000000013                0.0]
        [               0.0                0.0                0.0 1.0000000000000007]
        sage: T.parent()
        Full MatrixSpace of 4 by 4 dense matrices over Complex Double Field
        sage: all([T.zero_at(1.0e-12)[i,j] == 0 for i in range(4) for j in range(i)])
        True
        sage: (Q*T*Q.conjugate().transpose()-A).zero_at(1.0e-11)
        [0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0]

    Now totally over the reals. But with complex eigenvalues, the similar matrix may not be upper-triangular. But “at worst” there may be some \(2\times 2\) blocks on the diagonal which represent a pair of conjugate complex eigenvalues. These blocks will then just interrupt the zeros below the main diagonal. This example has a pair of these of the blocks.

        sage: A = matrix(RDF, 4, 4, [[1, 0, -3, -1],
        ....:                        [4, -16, -7, 0],
        ....:                        [1, 21, 1, -2],
        ....:                        [26, -1, -2, 1]])
        sage: Q, T = A.schur()
        sage: (Q*Q.conjugate().transpose())  # tol 1e-12
        [0.9999999999999994                0.0                0.0                0.0]
        [               0.0 1.0000000000000013                0.0                0.0]
        [               0.0                0.0 1.0000000000000004                0.0]
        [               0.0                0.0                0.0 1.0000000000000016]
        sage: all([T.zero_at(1.0e-12)[i,j] == 0 for i in range(4) for j in range(i)])
        False
        sage: all([T.zero_at(1.0e-12)[i,j] == 0 for i in range(4) for j in range(i-1)])
        True
        sage: (Q*T*Q.conjugate().transpose()-A).zero_at(1.0e-11)
        [0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0]
        sage: sorted(T[0:2,0:2].eigenvalues() + T[2:4,2:4].eigenvalues())
        [-5.710... - 8.382...*I, -5.710... + 8.382...*I, -0.789... - 2.336...*I, -0.789... + 2.336...*I]
        sage: sorted(A.eigenvalues())
        [-5.710... - 8.382...*I, -5.710... + 8.382...*I, -0.789... - 2.336...*I, -0.789... + 2.336...*I]
        sage: abs(A.norm()-T.norm()) < 1e-12
        True

    Starting with complex numbers and requesting a result over the reals will never happen.

        sage: A = matrix(CDF, 2, 2, [[2+I, -1+3*I], [5-4*I, 2-7*I]])
        sage: A.schur(base_ring=RDF)
        Traceback (most recent call last):
        ...
        TypeError: unable to convert input matrix over CDF to a matrix over RDF

    If theory predicts your matrix is real, but it contains some very small imaginary parts, you can specify the cutoff for “small” imaginary parts, then request the output as real matrices, and let the routine do the rest.

        sage: A = matrix(RDF, 2, 2, [1, 1, -1, 0]) + matrix(CDF, 2, 2, [1.0e-14*I]*4)
        sage: B = A.zero_at(1.0e-12)
        sage: B.parent()
        Full MatrixSpace of 2 by 2 dense matrices over Complex Double Field
        sage: Q, T = B.schur(RDF)
        sage: Q.parent()
        Full MatrixSpace of 2 by 2 dense matrices over Real Double Field
        sage: T.parent()
        Full MatrixSpace of 2 by 2 dense matrices over Real Double Field
        sage: Q.round(6)
        [ 0.707107  0.707107]
        [-0.707107  0.707107]
        sage: T.round(6)
        [ 0.5  1.5]
        [-0.5  0.5]
        sage: (Q*T*Q.conjugate().transpose()-B).zero_at(1.0e-11)
        [0.0 0.0]
        [0.0 0.0]

    A Hermitian matrix has real eigenvalues, so the similar matrix will be upper-triangular. Furthermore, a Hermitian matrix is diagonalizable with respect to an orthonormal basis, composed of eigenvectors of the matrix. Here that basis is the set of columns of the unitary matrix.

        sage: A = matrix(CDF, [[        52,   -9*I - 8,    6*I - 187,  -188*I + 2],
        ....:                  [   9*I - 8,         12,   -58*I + 59,   30*I + 42],
        ....:                  [-6*I - 187,  58*I + 59,         2677, 2264*I + 65],
        ....:                  [ 188*I + 2, -30*I + 42, -2264*I + 65,       2080]])
        sage: Q, T = A.schur()
        sage: T = T.zero_at(1.0e-12).change_ring(RDF)
        sage: T.round(6)
        [4680.13301        0.0        0.0        0.0]
        [       0.0 102.715967        0.0        0.0]
        [       0.0        0.0  35.039344        0.0]
        [       0.0        0.0        0.0    3.11168]
        sage: (Q*Q.conjugate().transpose()).zero_at(1e-12)  # tol 1e-12
        [1.0000000000000004                0.0                0.0                0.0]
        [               0.0 0.9999999999999989                0.0                0.0]
        [               0.0                0.0 1.0000000000000002                0.0]
        [               0.0                0.0                0.0 0.9999999999999992]
        sage: (Q*T*Q.conjugate().transpose()-A).zero_at(1.0e-11)
        [0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0]

    Similarly, a real symmetric matrix has only real eigenvalues, and there is an orthonormal basis composed of eigenvectors of the matrix.

        sage: A = matrix(RDF, [[ 1, -2, 5, -3],
        ....:                  [-2,  9, 1,  5],
        ....:                  [ 5,  1, 3 , 7],
        ....:                  [-3,  5, 7, -8]])
        sage: Q, T = A.schur()
        sage: Q.round(4)
        [-0.3027  -0.751   0.576 -0.1121]
        [  0.139 -0.3892 -0.2648  0.8713]
        [ 0.4361   0.359  0.7599  0.3217]
        [ -0.836  0.3945  0.1438  0.3533]
        sage: T = T.zero_at(10^-12)
        sage: all(abs(e) < 10^-4 for e in (T - diagonal_matrix(RDF, [-13.5698, -0.8508, 7.7664, 11.6542])).list())
        True
        sage: (Q*Q.transpose())  # tol 1e-12
        [0.9999999999999998                0.0                0.0                0.0]
        [               0.0                1.0                0.0                0.0]
        [               0.0                0.0 0.9999999999999998                0.0]
        [               0.0                0.0                0.0 0.9999999999999996]
        sage: (Q*T*Q.transpose()-A).zero_at(1.0e-11)
        [0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0]
        [0.0 0.0 0.0 0.0]

    The results are cached, both as a real factorization and also as a complex factorization. This means the returned matrices are immutable.

        sage: A = matrix(RDF, 2, 2, [[0, -1], [1, 0]])
        sage: Qr, Tr = A.schur(base_ring=RDF)
        sage: Qc, Tc = A.schur(base_ring=CDF)
        sage: all([M.is_immutable() for M in [Qr, Tr, Qc, Tc]])
        True
        sage: Tr.round(6) != Tc.round(6)
        True

    TESTS:

    The Schur factorization is only defined for square matrices.

        sage: A = matrix(RDF, 4, 5, range(20))
        sage: A.schur()
        Traceback (most recent call last):
        ...
        ValueError: Schur decomposition requires a square matrix, not a 4 x 5 matrix

    A base ring request is checked.

        sage: A = matrix(RDF, 3, range(9))
        sage: A.schur(base_ring=QQ)
        Traceback (most recent call last):
        ...
        ValueError: base ring of Schur decomposition matrices must be RDF or CDF, not Rational Field

    AUTHOR:

    - Rob Beezer (2011-03-31)
  `singular_values`(eps=None)

    Returns a sorted list of the singular values of the matrix.

    INPUT:

    - `eps` - default: `None` - the largest number which will be considered to be zero. May also be set to the string ‘auto’. See the discussion below.

    OUTPUT:

    A sorted list of the singular values of the matrix, which are the diagonal entries of the “S” matrix in the SVD decomposition. As such, the values are real and are returned as elements of `RDF`. The list is sorted with larger values first, and since theory predicts these values are always positive, for a rank-deficient matrix the list should end in zeros (but in practice may not). The length of the list is the minimum of the row count and column count for the matrix.

    The number of non-zero singular values will be the rank of the matrix. However, as a numerical matrix, it is impossible to control the difference between zero entries and very small non-zero entries. As an informed consumer it is up to you to use the output responsibly. We will do our best, and give you the tools to work with the output, but we cannot give you a guarantee.

    With `eps` set to `None` you will get the raw singular values and can manage them as you see fit. You may also set `eps` to any positive floating point value you wish. If you set `eps` to ‘auto’ this routine will compute a reasonable cutoff value, based on the size of the matrix, the largest singular value and the smallest nonzero value representable by the 53-bit precision values used. See the discussion at page 268 of [WATKINS].

    See the examples for a way to use the “verbose” facility to easily watch the zero cutoffs in action.

    ALGORITHM:

    The singular values come from the SVD decomposition computed by SciPy/NumPy.

    EXAMPLES:

    Singular values close to zero have trailing digits that may vary on different hardware. For exact matrices, the number of non-zero singular values will equal the rank of the matrix. So for some of the doctests we round the small singular values that ideally would be zero, to control the variability across hardware.

    This matrix has a determinant of one. A chain of two or three theorems implies the product of the singular values must also be one.

        sage: A = matrix(QQ, [[ 1,  0,  0,  0,  0,  1,  3],
        ....:                 [-2,  1,  1, -2,  0, -4,  0],
        ....:                 [ 1,  0,  1, -4, -6, -3,  7],
        ....:                 [-2,  2,  1,  1,  7,  1, -1],
        ....:                 [-1,  0, -1,  5,  8,  4, -6],
        ....:                 [ 4, -2, -2,  1, -3,  0,  8],
        ....:                 [-2,  1,  0,  2,  7,  3, -4]])
        sage: A.determinant()
        1
        sage: B = A.change_ring(RDF)
        sage: sv = B.singular_values(); sv  # tol 1e-12
        [20.523980658874265, 8.486837028536643, 5.86168134845073, 2.4429165899286978, 0.5831970144724045, 0.26933287286576313, 0.0025524488076110402]
        sage: prod(sv)  # tol 1e-12
        0.9999999999999525

    An exact matrix that is obviously not of full rank, and then a computation of the singular values after conversion to an approximate matrix.

        sage: A = matrix(QQ, [[1/3, 2/3, 11/3],
        ....:                 [2/3, 1/3,  7/3],
        ....:                 [2/3, 5/3, 27/3]])
        sage: A.rank()
        2
        sage: B = A.change_ring(CDF)
        sage: sv = B.singular_values()
        sage: sv[0:2]
        [10.1973039..., 0.487045871...]
        sage: sv[2] < 1e-14
        True

    A matrix of rank 3 over the complex numbers.

        sage: A = matrix(CDF, [[46*I - 28, -47*I - 50, 21*I + 51, -62*I - 782, 13*I + 22],
        ....:                  [35*I - 20, -32*I - 46, 18*I + 43, -57*I - 670, 7*I + 3],
        ....:                  [22*I - 13, -23*I - 23, 9*I + 24, -26*I - 347, 7*I + 13],
        ....:                  [-44*I + 23, 41*I + 57, -19*I - 54, 60*I + 757, -11*I - 9],
        ....:                  [30*I - 18, -30*I - 34, 14*I + 34, -42*I - 522, 8*I + 12]])
        sage: sv = A.singular_values()
        sage: sv[0:3]  # tol 1e-14
        [1440.7336659952966, 18.404403413369227, 6.839707797136151]
        sage: (10^-15 < sv[3] < 10^-13) or sv[3]
        True
        sage: (10^-16 < sv[4] < 10^-14) or sv[4]
        True

    A full-rank matrix that is ill-conditioned. We use this to illustrate ways of using the various possibilities for `eps`, including one that is ill-advised. Notice that the automatically computed cutoff gets this (difficult) example slightly wrong. This illustrates the impossibility of any automated process always getting this right. Use with caution and judgement.

        sage: entries = [1/(i+j+1) for i in range(12) for j in range(12)]
        sage: B = matrix(QQ, 12, 12, entries)
        sage: B.rank()
        12
        sage: A = B.change_ring(RDF)
        sage: A.condition() > 1.6e16 or A.condition()
        True

        sage: A.singular_values(eps=None)  # abs tol 7e-16
        [1.7953720595619975, 0.38027524595503703, 0.04473854875218107, 0.0037223122378911614, 0.0002330890890217751, 1.116335748323284e-05, 4.082376110397296e-07, 1.1228610675717613e-08, 2.2519645713496478e-10, 3.1113486853814003e-12, 2.6500422260778388e-14, 9.87312834948426e-17]
        sage: A.singular_values(eps='auto')  # abs tol 7e-16
        [1.7953720595619975, 0.38027524595503703, 0.04473854875218107, 0.0037223122378911614, 0.0002330890890217751, 1.116335748323284e-05, 4.082376110397296e-07, 1.1228610675717613e-08, 2.2519645713496478e-10, 3.1113486853814003e-12, 2.6500422260778388e-14, 0.0]
        sage: A.singular_values(eps=1e-4)  # abs tol 7e-16
        [1.7953720595619975, 0.38027524595503703, 0.04473854875218107, 0.0037223122378911614, 0.0002330890890217751, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]

    With Sage’s “verbose” facility, you can compactly see the cutoff at work. In any application of this routine, or those that build upon it, it would be a good idea to conduct this exercise on samples. We also test here that all the values are returned in \(RDF\) since singular values are always real.

        sage: A = matrix(CDF, 4, range(16))
        sage: set_verbose(1)
        sage: sv = A.singular_values(eps='auto'); sv
        verbose 1 (<module>) singular values,
        smallest-non-zero:cutoff:largest-zero,
        2.2766...:6.2421...e-14:...
        [35.13996365902..., 2.27661020871472..., 0.0, 0.0]
        sage: set_verbose(0)

        sage: all([s in RDF for s in sv])
        True

    TESTS:

    Bogus values of the `eps` keyword will be caught.

        sage: A.singular_values(eps='junk')
        Traceback (most recent call last):
        ...
        ValueError: could not convert string to float: junk

    REFERENCES:

    [WATKINS] | Watkins, David S. Fundamentals of Matrix Computations, Third Edition. Wiley, Hoboken, New Jersey, 2010.

    AUTHOR:

    - Rob Beezer - (2011-02-18)
  `solve_left`(b)

    Solve the vector equation `x*A = b` for a nonsingular `A`.

    INPUT:

    - `self` - a square matrix that is nonsigular (of full rank).
    - `b` - a vector of the correct size. Elements of the vector must coerce into the base ring of the coefficient matrix. In particular, if `b` has entries from `CDF` then `self` must have `CDF` as its base ring.

    OUTPUT:

    The unique solution `x` to the matrix equation `x*A = b`, as a vector over the same base ring as `self`.

    ALGORITHM:

    Uses the `solve()` routine from the SciPy `scipy.linalg` module, after taking the tranpose of the coefficient matrix.

    EXAMPLES:

    Over the reals.

        sage: A = matrix(RDF, 3,3, [1,2,5,7.6,2.3,1,1,2,-1]); A
        [ 1.0  2.0  5.0]
        [ 7.6  2.3  1.0]
        [ 1.0  2.0 -1.0]
        sage: b = vector(RDF,[1,2,3])
        sage: x = A.solve_left(b); x.zero_at(1e-18) # fix noisy zeroes
        (0.666666666..., 0.0, 0.333333333...)
        sage: x.parent()
        Vector space of dimension 3 over Real Double Field
        sage: x*A  # tol 1e-14
        (0.9999999999999999, 1.9999999999999998, 3.0)

    Over the complex numbers.

        sage: A = matrix(CDF, [[      0, -1 + 2*I,  1 - 3*I,        I],
        ....:                  [2 + 4*I, -2 + 3*I, -1 + 2*I,   -1 - I],
        ....:                  [  2 + I,    1 - I,       -1,        5],
        ....:                  [    3*I,   -1 - I,   -1 + I,   -3 + I]])
        sage: b = vector(CDF, [2 -3*I, 3, -2 + 3*I, 8])
        sage: x = A.solve_left(b); x
        (-1.55765124... - 0.644483985...*I, 0.183274021... + 0.286476868...*I, 0.270818505... + 0.246619217...*I, -1.69003558... - 0.828113879...*I)
        sage: x.parent()
        Vector space of dimension 4 over Complex Double Field
        sage: abs(x*A - b) < 1e-14
        True

    The vector of constants, `b`, can be given in a variety of forms, so long as it coerces to a vector over the same base ring as the coefficient matrix.

        sage: A=matrix(CDF, 5, [1/(i+j+1) for i in range(5) for j in range(5)])
        sage: A.solve_left([1]*5)  # tol 1e-11
        (5.0, -120.0, 630.0, -1120.0, 630.0)

    TESTS:

    A degenerate case.

        sage: A = matrix(RDF, 0, 0, [])
        sage: A.solve_left(vector(RDF,[]))
        ()

    The coefficent matrix must be square.

        sage: A = matrix(RDF, 2, 3, range(6))
        sage: b = vector(RDF, [1,2,3])
        sage: A.solve_left(b)
        Traceback (most recent call last):
        ...
        ValueError: coefficient matrix of a system over RDF/CDF must be square, not 2 x 3

    The coefficient matrix must be nonsingular.

        sage: A = matrix(RDF, 5, range(25))
        sage: b = vector(RDF, [1,2,3,4,5])
        sage: A.solve_left(b)
        Traceback (most recent call last):
        ...
        LinAlgError: singular matrix

    The vector of constants needs the correct degree.

        sage: A = matrix(RDF, 5, range(25))
        sage: b = vector(RDF, [1,2,3,4])
        sage: A.solve_left(b)
        Traceback (most recent call last):
        ...
        TypeError: vector of constants over Real Double Field incompatible with matrix over Real Double Field

    The vector of constants needs to be compatible with the base ring of the coefficient matrix.

        sage: F.<a> = FiniteField(27)
        sage: b = vector(F, [a,a,a,a,a])
        sage: A.solve_left(b)
        Traceback (most recent call last):
        ...
        TypeError: vector of constants over Finite Field in a of size 3^3 incompatible with matrix over Real Double Field

    With a coefficient matrix over `RDF`, a vector of constants over `CDF` can be accomodated by converting the base ring of the coefficient matrix.

        sage: A = matrix(RDF, 2, range(4))
        sage: b = vector(CDF, [1+I,2])
        sage: A.solve_left(b)
        Traceback (most recent call last):
        ...
        TypeError: vector of constants over Complex Double Field incompatible with matrix over Real Double Field

        sage: B = A.change_ring(CDF)
        sage: B.solve_left(b)
        (0.5 - 1.5*I, 0.5 + 0.5*I)

  `solve_left_LU`(b)

    Solve the equation \(A x = b\) using LU decomposition.

    Warning

    This function is broken. See trac 4932.

    INPUT:

    - self – an invertible matrix
    - b – a vector

    Note

    This method precomputes and stores the LU decomposition before solving. If many equations of the form Ax=b need to be solved for a singe matrix A, then this method should be used instead of solve. The first time this method is called it will compute the LU decomposition. If the matrix has not changed then subsequent calls will be very fast as the precomputed LU decomposition will be reused.

    EXAMPLES:

        sage: A = matrix(RDF, 3,3, [1,2,5,7.6,2.3,1,1,2,-1]); A
        [ 1.0  2.0  5.0]
        [ 7.6  2.3  1.0]
        [ 1.0  2.0 -1.0]
        sage: b = vector(RDF,[1,2,3])
        sage: x = A.solve_left_LU(b); x
        Traceback (most recent call last):
        ...
        NotImplementedError: this function is not finished (see trac 4932)

    TESTS:

    We test two degenerate cases:

        sage: A = matrix(RDF, 0, 3, [])
        sage: A.solve_left_LU(vector(RDF,[]))
        (0.0, 0.0, 0.0)
        sage: A = matrix(RDF, 3, 0, [])
        sage: A.solve_left_LU(vector(RDF,3, [1,2,3]))
        ()

  `solve_right`(b)

    Solve the vector equation `A*x = b` for a nonsingular `A`.

    INPUT:

    - `self` - a square matrix that is nonsigular (of full rank).
    - `b` - a vector of the correct size. Elements of the vector must coerce into the base ring of the coefficient matrix. In particular, if `b` has entries from `CDF` then `self` must have `CDF` as its base ring.

    OUTPUT:

    The unique solution `x` to the matrix equation `A*x = b`, as a vector over the same base ring as `self`.

    ALGORITHM:

    Uses the `solve()` routine from the SciPy `scipy.linalg` module.

    EXAMPLES:

    Over the reals.

        sage: A = matrix(RDF, 3,3, [1,2,5,7.6,2.3,1,1,2,-1]); A
        [ 1.0  2.0  5.0]
        [ 7.6  2.3  1.0]
        [ 1.0  2.0 -1.0]
        sage: b = vector(RDF,[1,2,3])
        sage: x = A.solve_right(b); x  # tol 1e-14
        (-0.1136950904392765, 1.3901808785529717, -0.33333333333333337)
        sage: x.parent()
        Vector space of dimension 3 over Real Double Field
        sage: A*x  # tol 1e-14
        (1.0, 1.9999999999999996, 3.0000000000000004)

    Over the complex numbers.

        sage: A = matrix(CDF, [[      0, -1 + 2*I,  1 - 3*I,        I],
        ....:                  [2 + 4*I, -2 + 3*I, -1 + 2*I,   -1 - I],
        ....:                  [  2 + I,    1 - I,       -1,        5],
        ....:                  [    3*I,   -1 - I,   -1 + I,   -3 + I]])
        sage: b = vector(CDF, [2 -3*I, 3, -2 + 3*I, 8])
        sage: x = A.solve_right(b); x
        (1.96841637... - 1.07606761...*I, -0.614323843... + 1.68416370...*I, 0.0733985765... + 1.73487544...*I, -1.6018683... + 0.524021352...*I)
        sage: x.parent()
        Vector space of dimension 4 over Complex Double Field
        sage: abs(A*x - b) < 1e-14
        True

    The vector of constants, `b`, can be given in a variety of forms, so long as it coerces to a vector over the same base ring as the coefficient matrix.

        sage: A=matrix(CDF, 5, [1/(i+j+1) for i in range(5) for j in range(5)])
        sage: A.solve_right([1]*5)  # tol 1e-11
        (5.0, -120.0, 630.0, -1120.0, 630.0)

    TESTS:

    A degenerate case.

        sage: A = matrix(RDF, 0, 0, [])
        sage: A.solve_right(vector(RDF,[]))
        ()

    The coefficent matrix must be square.

        sage: A = matrix(RDF, 2, 3, range(6))
        sage: b = vector(RDF, [1,2,3])
        sage: A.solve_right(b)
        Traceback (most recent call last):
        ...
        ValueError: coefficient matrix of a system over RDF/CDF must be square, not 2 x 3

    The coefficient matrix must be nonsingular.

        sage: A = matrix(RDF, 5, range(25))
        sage: b = vector(RDF, [1,2,3,4,5])
        sage: A.solve_right(b)
        Traceback (most recent call last):
        ...
        LinAlgError: singular matrix

    The vector of constants needs the correct degree.

        sage: A = matrix(RDF, 5, range(25))
        sage: b = vector(RDF, [1,2,3,4])
        sage: A.solve_right(b)
        Traceback (most recent call last):
        ...
        TypeError: vector of constants over Real Double Field incompatible with matrix over Real Double Field

    The vector of constants needs to be compatible with the base ring of the coefficient matrix.

        sage: F.<a> = FiniteField(27)
        sage: b = vector(F, [a,a,a,a,a])
        sage: A.solve_right(b)
        Traceback (most recent call last):
        ...
        TypeError: vector of constants over Finite Field in a of size 3^3 incompatible with matrix over Real Double Field

    With a coefficient matrix over `RDF`, a vector of constants over `CDF` can be accomodated by converting the base ring of the coefficient matrix.

        sage: A = matrix(RDF, 2, range(4))
        sage: b = vector(CDF, [1+I,2])
        sage: A.solve_right(b)
        Traceback (most recent call last):
        ...
        TypeError: vector of constants over Complex Double Field incompatible with matrix over Real Double Field

        sage: B = A.change_ring(CDF)
        sage: B.solve_right(b)
        (-0.5 - 1.5*I, 1.0 + 1.0*I)

  `transpose`()

    Return the transpose of this matrix, without changing self.

    EXAMPLES:

        sage: m = matrix(RDF,2,3,range(6)); m
        [0.0 1.0 2.0]
        [3.0 4.0 5.0]
        sage: m2 = m.transpose()
        sage: m[0,0] = 2
        sage: m2           #note that m2 hasn't changed
        [0.0 3.0]
        [1.0 4.0]
        [2.0 5.0]

    `.T` is a convenient shortcut for the transpose:

        sage: m.T
        [2.0 3.0]
        [1.0 4.0]
        [2.0 5.0]

        sage: m = matrix(RDF,0,3); m
        []
        sage: m.transpose()
        []
        sage: m.transpose().parent()
        Full MatrixSpace of 3 by 0 dense matrices over Real Double Field

  `zero_at`(eps)

    Returns a copy of the matrix where elements smaller than or equal to `eps` are replaced with zeroes. For complex matrices, the real and imaginary parts are considered individually.

    This is useful for modifying output from algorithms which have large relative errors when producing zero elements, e.g. to create reliable doctests.

    INPUT:

    - `eps` - Cutoff value

    OUTPUT:

    A modified copy of the matrix.

    EXAMPLES:

        sage: a=matrix([[1, 1e-4r, 1+1e-100jr], [1e-8+3j, 0, 1e-58r]])
        sage: a
        [           1.0         0.0001 1.0 + 1e-100*I]
        [ 1e-08 + 3.0*I            0.0          1e-58]
        sage: a.zero_at(1e-50)
        [          1.0        0.0001           1.0]
        [1e-08 + 3.0*I           0.0           0.0]
        sage: a.zero_at(1e-4)
        [  1.0   0.0   1.0]
        [3.0*I   0.0   0.0]

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