Product of diagonal and symmetric positive definite matrix. Let $C$ be an $n \times n$ diagonal matrix with positive diagonal entries, and let $G$ be an $n \times n$ symmetric positive definite matrix.  What can we say about $CG + GC$?  For example, is it non-singular?  Is it positive definite?  What restrictions on $C$ and/or $G$ would guarantee that $CG + GC$ is non-singular?

Let $M=CG+GC$. The eigenvalues of $M$ are the same with those of $C^{-1/2}MC^{1/2} = C^{1/2}GC^{-1/2}+C^{-1/2}GC^{1/2}$. Since $C^{1/2}GC^{-1/2}$ and $C^{-1/2}GC^{1/2}$ are symmetric and positive definite, $C^{-1/2}MC^{1/2}$ is also positive definite. Thus, all eigenvalues of $M$ are positive, which together with the fact that $M$ is symmetric imply that $M$ is positive definite.
