Let $A_1$ and $A_2$ be positive semi-definite matrices such that Tr$(A_1) \leq$ Tr$(A_2)$. Let $B$ be another positive semi-definite matrix. Is it true that Tr$(A_1B) \leq$ Tr$(A_2B)$?
2026-03-27 06:08:48.1774591728
Trace inequality on the product of positive semi-definite matrices
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No. Set
$$A_1=\begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}$$ $$A_2=\begin{bmatrix} 10 & 0 \\ 0 & 1 \end{bmatrix}$$ $$B=\begin{bmatrix} 0.1 & 0 \\ 0 & 2 \end{bmatrix}$$