Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
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 a ValueError
results.
ALGORITHM:
Whether or not the matrix is positive definite is checked first in every case. This is accomplished with
an indefinite factorization (see indefinite_factorization()) which caches its result. This algorithm
is of an order n 3 /3. If the matrix is positive definite, this computation always succeeds, using just
field operations. The transistion to a Cholesky decomposition “only” requires computing square roots of
the positive (real) entries of the diagonal matrix produced in the indefinite factorization. Hence, there is no
real penalty in the positive definite check (here, or prior to calling this routine), but a field extension with
square roots may not be implemented in all reasonable cases.
EXAMPLES:
This simple example has a result with entries that remain in the field of rational numbers.
sage: A = matrix(QQ, [[ 4, -2, 4, 2],
... [-2, 10, -2, -7],
... [ 4, -2, 8, 4],
... [ 2, -7, 4, 7]])
sage: A.is_symmetric()
True
sage: L = A.cholesky()
sage: L
[ 2 0 0 0]
[-1 3 0 0]
[ 2 0 2 0]
[ 1 -2 1 1]
sage: L.parent()
Full MatrixSpace of 4 by 4 dense matrices over Rational Field
sage: L*L.transpose() == A
True
This seemingly simple example requires first moving to the rational numbers for field operations, and then
square roots necessitate that the result has entries in the field of algebraic numbers.
sage: A = matrix(ZZ, [[ 78, -30, -37, -2],
... [-30, 102, 179, -18],
... [-37, 179, 326, -38],
... [ -2, -18, -38, 15]])
sage: A.is_symmetric()
True
sage: L = A.cholesky()
sage: L
[ 8.83176086632785 0 0 0]
[ -3.396831102433787 9.51112708681461 0 0]
[ -4.189425026335004 17.32383862241232 2.886751345948129 0]
[-0.2264554068289192 -1.973397116652010 -1.649572197684645 2.886751345948129]
sage: L.parent()
Full MatrixSpace of 4 by 4 dense matrices over Algebraic Field
sage: L*L.transpose() == A
True
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