Can someone just tell me if this proof works? As a quick word of explanation since most of my posts will probably be like this - I'm studying math on my own and am without a professor or anyone else to check things over. It would be nice to think that I'd know when a proof doesn't work, but I know I miss things sometimes.

"Let be a self-adjoint linear operator on an n-dimensional vector space , and let , where is an orthonormal basis for . Prove: is positive definite if and only if all of its eigenvalues are positive."

I know I got the forward direction, so here's the other direction.

Since T is self-adjoint, there exists an orthonormal basis (call it ) for consisting of eigenvectors of , which implies that is diagonalizable - that is, there exists an invertible matrix such that is a diagonal matrix. In particular, the entries of are the eigenvalues of . Letting be the entries of and the -th component of , we have , and so for all nonzero and is positive definite.