Re: Cholesky factors of matrices which are not positive-definite

On 2006-08-30 14:18:18 -0300, "Murphy O'Brien" <murphyobrien@xxxxxxxxx> said:

I have heard it said regarding Cholesky factorisation, that :

"A symmetric matrix P is positive definite if and only if it has a
factorization P= L*L.' with a nonsingular lower triangular matrix."

Whet L.' is the transpose of L

P and L will be real matrices if you read the rest of the surrounding
material.

The extension to complex values will be for P Hermitian.



But that seems to only be true if one doesn't allow complex numbers.
e.g. the following matrix


P
1 1 -1
1 2 3
-1 3 6


is not positive definite becuase it has eigenvalues


-0.6977 2.0658 7.6319


but has a factorization L


1 0 0
1 1 0
-1 4 - 3.31662479035540i


where P=L*L.'


Where is the problem?

Quoting from incomplete context.

The quoted sentence above, or my interpretation of it, or my proof that
it's wrong.

Murphy


.

Relevant Pages