Eigenvalues of matrix product $AXA$

Question

Given a diagonal matrix $A = \operatorname{diag}(a_{1}, a_{2}, \cdots, a_{n})\in\mathbb{R}^{n\times n}$ with $a_{1}>a_{2}>\cdots>a_{n}>0$ and a positive definite matrix $X\in\mathbb{R}^{n\times n}$ with eigenvalues $x_{1}>x_{2}>\cdots>x_{n}>0$.

I was wondering that could we obtain the eigenvalue inequality like $$\lambda_{i}(AXA)\leq a_{i}^2 x_{i}$$ where $\lambda_{i}$ denotes the $i$-th largest eigenvalue. If not, what is the correct one?

Answer 1

The inequality in your case will be $$\lambda_{\max}(AXA)\leq a_1^2x_1,$$

which follows from the fact that for real positive definite matrices $M,N$,

$$\lambda_\text{max}(MN)\leq \lambda_\text{max}(M)\lambda_\text{max}(N),$$

as discussed here.