How to check if a matrix is positive definite

I don't think there is a nice answer for matrices in general. Most often we care about positive definite matrices for Hermitian matrices, so a lot is known in this case.

The one I always have in mind is that a Hermitian matrix is positive definite iff its eigenvalues are all positive.

Glancing at the wiki article on this alerted me to something I had not known, Sylvester's criterion which says that you can use determinants to test (a Hermitian matrix) for positive definiteness by checking to see if all the square submatrices whose upper left corner is the $(1,1)$ entry have positive determinant.

Sorry if this is repeating things you already know, but it's the most useful information I can provide. Good luck!