Product of diagonal and positive definite matrix.

Let $C$ be an $n \times n$ diagonal matrix with positive diagonal entries, and let $G$ be an $n \times n$ ymmetric 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?

• Take unit matrix as diagonal matrix and see what happens – Archis Welankar Jan 10 '16 at 6:01

Clearly $S=CG+GC$ is real symmetric. There are instances s.t. $S$ is not $>0$ or s.t. $\det(S)=0$. Note that the eigenvalues of $CG$ and $GC$ are $>0$.
Now, if $GC=CG$, then $GC$ is symmetric $>0$; of course $S$ is also symmetric $>0$.
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.
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.