Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
[0.0 1.0 0.0]
[0.0 0.0 1.0]
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()
[1.0 2.0]
[3.0 4.0]
[5.0 6.0]
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 ∗
where L ∗ 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.
356 Chapter 19. Dense matrices using a NumPy backend.