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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$(A_2B)$?

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    $\begingroup$ If $A_2 - A_1$ is positive semidefinite, then you do get the desired inequality $\endgroup$ – Omnomnomnom Jan 26 '17 at 13:14
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No. Set

$$A_1=\begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}$$ $$A_2=\begin{bmatrix} 10 & 0 \\ 0 & 1 \end{bmatrix}$$ $$B=\begin{bmatrix} 0.1 & 0 \\ 0 & 2 \end{bmatrix}$$

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