# Simple sufficient conditions for matrix positive semidefiniteness?

I need to show that a complicated $n \times n$ Hermitian matrix is positive semidefinite. I'm wondering if there are simple sufficient conditions that can be used to show this. For instance, if a matrix with positive diagonal is diagonally dominant, then it is positive semidefinite. Other conditions like this would be helpful.

• It is pos. semidefinite iff all its eigenvalues are (real) non-negative, iff all its principal minors are non-negative. – DonAntonio May 26 '17 at 21:28
• Right, I guess I should have been more precise with my definition of "simple". Computing the eigenvalues or the determinants of large matrices is not very simple. – user_lambda May 26 '17 at 23:33
• Anything else would imply, imo, either much more information regarding a particular matrix, or else magic... – DonAntonio May 26 '17 at 23:39