I think the title is self-explanatory. I would like to have a small proof supporting your statement. $\pmb{x}$ and $\pmb{y}$ denote vectors and $A$ denote a square positive definite matrix.
2026-03-25 06:04:43.1774418683
Is $\pmb{x}<\pmb{y}$ implies $A\pmb{x}<A\pmb{y}$ when $A$ is positive definite?
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No, consider $A=\begin{bmatrix}1&-1\\-1&3\end{bmatrix}$ and $v=\begin{bmatrix}1\\1\end{bmatrix}, w=2v$.