Let $K$ be a symmetric, positive definite matrix and $S$ be a diagonal matrix such that all entries on its diagonal are greater than $1$. I am trying to prove the following relation.
$$ \| S K S - K \| \leq \|K\| \left( \| S \|^2 - 1 \right),$$
where $\| \cdot \|$ denotes the spectral norm of a matrix.
I feel that intuitively the relation should hold but I haven't been able to prove it. Any hints or solutions (or even a counterexample) would be appreciated. Thanks!
It is true. Since $S\succeq I$, we have $\|S-I\|=\|S\|-1$. Therefore $$ \begin{aligned} \|SKS-K\| &\le\|SKS-KS\|+\|KS-K\|\\ &\le\|S-I\|\|K\|\|S\|+\|K\|\|S-I\|\\ &=(\|S\|-1)\|K\|\|S\|+\|K\|(\|S\|-1)\\ &=\|K\|(\|S\|^2-1). \end{aligned} $$ Note that $K$ doesn't need to be positive definite. It can be a general square matrix. $S$ also doesn't need to be a diagonal matrix. As long as $S\succeq I$, the proof above remains valid.