# A monotonicity property of eigenvalues

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.

• @FedorPetrov. $D$ is diagonal. Mar 12, 2021 at 7:29

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