[Math] Checking if matrix $A$ is positive definite via Cholesky decomposition

Question

How can we show that a matrix $A$ is not a positive definite matrix using the Cholesky decomposition?

If we are not able to complete the algorithm and we cannot factor the matrix with a Cholesky decomposition does it then mean that the matrix is not positiv definite?

Or is there an other way to check whether the matrix is positiv definite or not?

Answer 1

Cholesky decomposition will fail only when the matrix is not symmetric positive semi definite. Thus, if the algorithm doesn’t work, then you know your matrix is not symmetric positive semidefinite.