If $\operatorname{Tr}(A) < \operatorname{Tr}(B)$, is it fair to say that $\operatorname{Tr}(AC) < \operatorname{Tr}(BC)$? All of $A$, $B$ and $C$ are positive definite matrices.
2026-04-01 04:58:12.1775019492
Matrix Trace Inequality
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Let $$A=\begin{pmatrix} 1 & 0 \\ 0 & 1.1 \end{pmatrix}, B=\begin{pmatrix} 1.2 & 0 \\ 0 & 1 \end{pmatrix},C=\begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}.$$
Can you conclude anything?