TITLE: Trace inequality on the product of positive semi-definite matrices
QUESTION [5 upvotes]: 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$(A_2B)$?

REPLY [6 votes]: No. Set
$$A_1=\begin{bmatrix} 
3 & 0 \\
0 & 3 
\end{bmatrix}$$
$$A_2=\begin{bmatrix} 
10 & 0 \\
0 & 1 
\end{bmatrix}$$
$$B=\begin{bmatrix} 
0.1 & 0 \\
0 & 2 
\end{bmatrix}$$