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?

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*Notation: $\Re_+$ denotes the set of positive scalar real numbers

Let diagonal positive definite matrix be: $C=diag(c_1, c_2, \cdots, c_n)$ with $c_i\in\Re_+$, for $i=1,2,\cdots,n$. Also, let the positive definite matrix $G$ be denoted as
$G=\left[\begin{matrix} g_{11} & g_{12} & \cdots & g_{1n} \\ \vdots & \vdots & \vdots & \vdots \\ g_{n1} & g_{n2} & \cdots & g_{nn} \end{matrix}\right]$
Since $G$ is a symmetric positive definite matrix, all the leading principal minors $\pi_i$ ($i=1,2,\cdots,n$) are positive from Sylvester's criterion. That is,
$\pi_1 = g_{11}>0, \quad \pi_2 = g_{11}g_{22}-g_{12}^2 > 0, \quad \cdots$
It is straightforward to show that the leading principal minors of the product of $CG$ ($\pi_i^\prime$) are given by
$\pi^\prime_n = \left(\prod_{i=1}^{n}c_i\right)\pi_n > 0$
Hence, the product $CG$ is a positive definite matrix, but not necessarily symmetric.
Note that $C = C^T$ and $G=G^T$ are symmetric matrices by their definitions. Thus, $(CG)^T=G^TC^T=GC$. Then, the product $CG$ is symmetric.
Combining these results, one can conclude that the product of $CG$ is a symmetric positive definite matrix provided that $C$ is a diagonal matrix with positive elements and $G$ is a symmetric positive definite matrix.
