Is there any sufficient or necessary conditions for a matrix to have LU factorization?

Question

I have learnt that the matrix has to be a real, symmetric and a positive definite matrix to find it's LU factorization by Cholesky's method. Is there any conditions for Doolittle's method and Crout's method?

Answer 1

As described here, you need to have all the leading principal minors be non-singular. If you allow permutations, then the matrix just needs to be invertible.