Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: [A[:i,:i].determinant() for i in range(1,A.nrows()+1)]
[21, 27, 0, 0]
sage: A.cholesky()
Traceback (most recent call last):
...
ValueError: matrix is not positive definite,
so cannot compute Cholesky decomposition
Even in light of the above, you can sometimes get lucky and arrive at a situation where a particular matrix
has a Cholesky decomposition when the general characteristics of the matrix suggest this routine would
fail. In this example, the indefinite factorization produces a diagonal matrix whose elements from the finite
field convert naturally to positive integers and are also perfect squares.
sage: F. = FiniteField(5^3)
sage: A = matrix(F, [[ 4, 2*a^2 + 3, 4*a + 1],
... [ 2*a^2 + 3, 2*a + 2, 4*a^2 + 4*a + 4],
... [ 4*a + 1, 4*a^2 + 4*a + 4, a^2 + 4*a]])
sage: A.is_symmetric()
True
sage: L = A.cholesky()
sage: L*L.transpose() == A
True
TESTS:
This verifies that trac ticket #11274 is resolved.
sage: E = matrix(QQ, [[2, 1], [1, 1]])
sage: E.is_symmetric()
True
sage: E.eigenvalues()
[0.38..., 2.61...]
sage: E.det()
1
sage: E.cholesky()
[ 1.414213562373095 0]
[0.7071067811865475 0.7071067811865475]
AUTHOR:
•Rob Beezer (2012-05-27)
cholesky_decomposition()
Return the Cholesky decomposition of self.
Warning: cholesky_decomposition() is deprecated, please use cholesky() instead.
The computed decomposition is cached and returned on subsequent calls. Methods such as
solve_left() may also take advantage of the cached decomposition depending on the exact implementation.
INPUT:
The input matrix must be:
•real, symmetric, and positive definite; or
•complex, Hermitian, and positive definite.
If not, a ValueError exception will be raised.
138 Chapter 7. Base class for matrices, part 2