Let $A\in \mathbb{S}^n$ be PSD and let $S\in \mathbb{R}^{n\times n}$ be nonsingular. How to prove that for any $k\in \{1,\cdots,n\}$, $$\lambda_{min}(S^TS)\lambda_k(A)\le\lambda(S^TAS)\le\lambda_{max}(S^TS)\lambda_k(A)$$ It seems that Weyl inequality can be used. But I just cannot figure it out.
2026-03-26 12:32:17.1774528337
How tot prove the inequality: $\lambda_{min}(S^TS)\lambda_k(A)\le\lambda(S^TAS)\le\lambda_{max}(S^TS)\lambda_k(A)$
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