From: "R.G. Vickson"
Subject: Re: Question about Cholesky LLt vs. LDLt decomposition
Date: Fri, 01 Oct 1999 17:18:35 -0400
Newsgroups: sci.math
Keywords: Matrices which cannot be Cholesky-factored
John Henckel wrote:
> Cholesky's decomposition is a method for factoring matrices.
>
> I can see that the LLt (lower matrix times its transpose) decomposition
> is only valid for SPD (symmetric positive definite) matrices.
>
> The "modified Cholesky's method", LDLt, has a extra diagonal matrix
> factor. Several texts say this is simply a way to avoid the square roots
> (presumably because they are expensive to compute).
>
> However, it seems to me that the LDLt decomposition is much more
> powerful than LLt decomposition, because the LDLt decomposition (with
> pivoting) can factor ANY symmetric (non-singular) matrix, not just SPD.
I think this is false: some indefinite (symmetric, real, etc) matrices
cannot be factored at all int LDLt form. Some indefinite ones CAN BE
factored, but with negative numbers in the diagonal of D, but some cannot.
Simple example: the matrix
[0 1]
[1 0]
is indefinite and does not factor.
>
>
> Can someone please confirm this?
>
> Thanks in advance! Please send email.