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 T be a self-adjoint linear operator on an n-dimensional vector space V, and let A=[T]_\beta, where \beta is an orthonormal basis for V. Prove: T 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 \beta) for V consisting of eigenvectors of T, which implies that T is diagonalizable - that is, there exists an invertible matrix P such that D=P^{-1}[T]_\beta P is a diagonal matrix. In particular, the entries of D are the eigenvalues of T. Letting d_{ii} be the entries of D and x_i the i-th component of x, we have \displaystyle\sum_i d_{ii}x_i^2 > 0, and so \langle T(x), x \rangle > 0 for all nonzero x and T is positive definite.