# How to prove that a symmetric matrix is positive semidefinite?

I have a symmetric matrix whose diagonals are positive. I need to prove that this matrix is positive semidefinite.

The matrix is made up of a bunch of constants and I tried getting the eigenvalues using Maple and it was a mess. I also tried doing something I found online How to check if a symmetric $4\times4$ matrix is positive semi-definite?. I tried doing Robert Israel's answer and it ended up being a mess. Is there an easier way to prove positive semidefinite?

• How large is your matrix? Perhaps you can post it here. I would venture to guess that you may be able to apply the Gershgorin Circle Theorem. This is noted in Calle's answer in the question you linked to. – EuYu Nov 28 '12 at 4:21
• Symmetric matrices with positive diagonals aren't always positive. (E.g., $\begin{bmatrix}1&2\\2&1\end{bmatrix}$.) What is your matrix? – Jonas Meyer Nov 28 '12 at 4:21
• If your matrix is not too large, see this : math.stackexchange.com/questions/40849/… – Bhargav Nov 28 '12 at 4:22
• @b555: That is the question user972276 links to above. – Jonas Meyer Nov 28 '12 at 4:23
• oops my bad, sry .. had read the q i linked just a few days abck and couldnt read this q fully >.< – Bhargav Nov 28 '12 at 4:26