Why does the Cholesky factorization requires the matrix
A to be positive definite? What happens when we factorize non-positive definite matrix?
Let's assume that we have a matrix
A' that is not positive definite (so at least one leading principal minor is negative). Can one prove that there is no
L such as
A' = LL*? If not, wouldn't the positive definite criteria remove some of the matrices that could be potentially decomposed?
We could also put this question in the form of a demonstration for the next statement:
For any square matrix L, the product LL* is a positive definite matrix.