Positive- definiteness of factors of Cholesky factorization

Question

If A is a symmetric positive-definite n×n matrix, then is the lower triangular n×n matrix L positive-definite when A=LL* using Cholesky factorization?

Answer 1

In many cases that I have tested, the Cholesky factor is definite positive.

Nevertheless, there are cases where it is not. Here is a simple counterexample :

$$\underbrace{\begin{pmatrix}-2&1\end{pmatrix}}_{X^T}\underbrace{\begin{pmatrix}1 &3\\ 0 & 1\end{pmatrix}}_C\underbrace{\begin{pmatrix}-2\\ \ \ \ 1\end{pmatrix}}_X=-1,$$

$C$ being the Cholesky factor of

$$\begin{pmatrix}1& \ \ 3\\ 3&10\end{pmatrix}.$$

Remarks :

1) 99.99% of studied positive definite matrices are symmetrical which is not the case for Cholesky factors.

2) One can even say that it isn't natural to have a definite matrix which is not symmetric; indeed, whereas $X^TAX$ doesn't have "problems", the associated bilinear form $X^TAY$ has "problems" : it is not symmetric, i.e., in general $X^TAY \ne Y^TAX$ when $A^T \ne A$ !

3) Nevertheless, there are some studies on definite positive matrices that aren't symmetrical, this article.