Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
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(1.0e-12)
[1.0 0.0 0.0 0.0]
[0.0 1.0 0.0 0.0]
[0.0 0.0 1.0 0.0]
[0.0 0.0 0.0 1.0]
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],
385