I'm reading a paper that makes the following claim that I don't understand.
Let $\Sigma, K\in\mathbb{R}^{n,n}$, where $\Sigma$ is a diagonal matrix with only non-negative real entries. Then the claim says that
$$\max_{i\in\{1,\cdots,n\}}Re(\lambda_i(K))\leq 0$$
implies
$$\max_{i\in\{1,\cdots,n\}}Re(\lambda_i(\Sigma K))\leq 0,$$
where $\lambda_i(\cdot)$ denotes the $i^{th}$ eigenvalue of the respective matrix. There are no other requirements to either $\Sigma$ or $K$ beyond the ones mentioned.
Does anybody know why this is true?
For reference, the paper claiming this is this paper in equations (3.6) to (3.8).
As stated, the result is not true. For instance take $$ K=\begin{bmatrix} 2&-5\\ 5&-10\end{bmatrix},\ \ \ \Sigma=\begin{bmatrix}1&0\\0&0\end{bmatrix}. $$ The the eigenvalues of $K$ are negative, but $2$ is an eigenvalue of $\Sigma K$.
A small variation of the example, where $\Sigma$ is invertible, is to take
$$ \Sigma=\begin{bmatrix}1&0\\0&1/10\end{bmatrix}. $$ Then $\Sigma K$ has both eigenvalues with real part $+1/2$.