When is a matrix positive semi-definite?

JK423

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

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

micromass

You'll want the matrix to be Hermitian as well (or normal).

AlephZero

Complex symmetric (not hermitian!) matrices do occur in modelling processes which don't conserve energy, but then the concept of "positive semidefinite" isn't very meaningful. Ths signs of the real parts of the eignenvalues is usually more interesting physically - i.e. does the energy of the system increase or decrease.