Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
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()
[ 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]
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()
[1.0 2.0 3.0]
[4.0 5.0 6.0]
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()).zero_at(1e-15)
[1.0 0.0]
[0.0 1.0]
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()).zero_at(1e-15)
[1.0 0.0 0.0]
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