Eigenvalues of matrix product with a diagonal matrix

Question

I have the following problem: Suppose we have a nonnegative diagonal matrix $A \in \mathbb{R}_+^{n\times n}$ and a matrix $B \in \mathbb{R}^{n\times n}$ with $$Re(\lambda_i(B)) \leq 0,\; i=1,\dots,n$$

where $\lambda_i(B)$ denotes the $i$-th eigenvalue of $B$ and $Re(\cdot)$ denotes the real part.

Is it possible to show that $Re(\lambda_i(AB))\leq 0$ for $i=1,\dots,n$?

I found similar questions but with different conditions on $A$ and $B$:

• Eigenvalues of Matrix Product.

• If $A\ge0$ and $B\le 0$, are the eigenvalues of $AB$ non-positive?

Any help is appreciated!

Answer 1

A friend of mine found a simple counterexample:

$A=\begin{bmatrix}4 & 0\\0 & 1\end{bmatrix}, B=\begin{bmatrix}1 & -2\\2 & -2\end{bmatrix}$

For $B$ we have $Re(\lambda_{1}(B)) = Re(\lambda_{2}(B)) =-\frac{1}{2}$ and for $AB = \begin{bmatrix}4 & -8\\2 & -2\end{bmatrix}$ it is $Re(\lambda_{1}(AB)) = Re(\lambda_{2}(AB)) = 1$.