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?

Note that $G$ is symmetric Hence $G^T=G$. Also, it is trivial that the diagonal matrix $C$ is a symmetric matrix, therefore $C^T =C$. Let $M:=CG+GC$ Then we have  $$M^T=(CG+GC)^T = G^T C^T + C^T G^T = (GC+CG)=M.$$
Hence $M$ is symmetric. Obviously, $M$  is positive definite as multiplication and addition of positive definite matrices are positive definite. 
