Product of diagonal and symmetric positive definite matrix.

Question

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?

Answer 1

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 $>0$.

Answer 2

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.

Answer 3

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.

Answer 4

• 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.