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

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.