## When is a matrix positive semi-definite?

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

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 Recognitions: Gold Member Science Advisor Staff Emeritus 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

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 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).

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