Let $A \in S^{n}\_{+}$ be a positive semi-definite matrix and $D \in S^{n}\_{+}$ a diagonal matrix with all the diagonal entries no smaller than one, i.e., $D\_{ii} \geq 1$ for all $i \leq n$.


I wonder whether the transformation $DAD$ will scale up the eigenvalues? i.e., let $\lambda\_i(M)$ denote the $i$-th largest eigenvalue of a matrix $M$; is it always true that
$\lambda\_i(DAD) \geq \lambda\_i(A) \geq 0$ for all $i \leq n$?


I feel it is intuitive but haven't formally proved/disproved it yet. Forgive me if it is a commonly-known or obvious result.



A monotonicity property of eigenvalues
**Yes**.
Write the eigenvalues as $\max\min$ (or as $\min\max$) of Rayleigh quotient. Then use the fact that $\|Dx\|\ge\|x\|$.


