Prove that if $S = S^T$ is symmetric and non-singular, then $S^2$ is positive definite.
My attempt:
Suppose $S$ is an $m\times n$ symmetric matrix with linearly independent columns, and suppose $q(x) > 0$, then the matrix $q(x) = \mathbf{x}^\mathrm{T}S\mathbf{x}$ is a positive definite $n\times n$ matrix thus $S^2$ is also positive definite. Am I right, or completely off?