Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: all(abs(e) < 10^-4 for e in (T - diagonal_matrix(RDF, [-13.5698, -0.8508, 7.7664, 11.6
... [ 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)
True
sage: (Q*Q.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.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:
386 Chapter 19. Dense matrices using a NumPy backend.