# When is a matrix positive semi-definite?

by JK423
Tags: matrix, positive, semidefinite
 P: 381 Hello people, Im working on a project and this problem came up: I have a symmetric matrix whose elements are complex variables, and i know that this matrix is positive semi-definite. I have to derive a criterion for the matrix's elements, so that if it's satisfied by them then the matrix will be positive semi-definite. Any idea on how to do that? For example, a positive semi-definite matrix has to satisfy some relation that i can use? Maybe its eigenvalues must be non-negative? I'd really need your help, thanks a lot! John
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,495 Yes, a matrix is "positive semi-definite" if and only if all of its eigenvalues are non-negative. You might want to look at this: http://en.wikipedia.org/wiki/Positive-definite_matrix
Mentor
P: 18,231
 Quote by HallsofIvy Yes, a matrix is "positive semi-definite" if and only if all of its eigenvalues are non-negative. You might want to look at this: http://en.wikipedia.org/wiki/Positive-definite_matrix
You'll want the matrix to be Hermitian as well (or normal).

Engineering