Given that $A=LU$ where $L$ and $U$ are (known) lower and upper triangular matrices, is there any simple way to determine whether $A$ is positive definite?
I have been using this algorithm on Wikipedia to compute LU decompositions.
The comment section says:
INPUT: A ... a square matrix having dimension N
OUTPUT: Matrix A is changed, it contains a copy of both matrices L-E and U as A=(L-E)+U such that PA=LU.
I'm not sure what exactly 'E' refers to here, but I understand that 'L' refers to a lower triangular matrix, and 'U' to an upper triangular matrix, and I understand that the algorithm is letting these two cohabit the same square matrix.
I seek to determine whether the original matrix $A$ is positive definite. I have a feeling that $A$ is positive definite only if $A$ is symmetric and (after running the algorithm) all of the diagonal elements are positive. Is this correct? (Or is there some other way to determine whether $A$ is positive definite, from its LU decomposition?
If $A$ is a symmetric ... positive definite matrix, we can arrange matters so that $U$ is the ... transpose of $L$. That is, we can write $A$ as
$$A = L L^T$$
This decomposition is called the Cholesky decomposition.
Moreover I have read somewhere that a matrix is positive definite if and only if its Cholesky decomposition exists. But I don't know how to put all this together (e.g. what "arrange matters" above means exactly) to determine from the LU decomposition whether the matrix is positive definite.