Consider real matrix-vector multiplication. I am just curious whether there exists a $p\times p$ matrix $A$ such that if $x$ is a $p\times 1$ real vector whose entry is in ascending order, i.e., $x_1 \leq x_2 \leq \cdots \leq x_p$, then $Ax$ is still in ascending order. That is, $(Ax)_1 \leq \cdots \leq (Ax)_p$. If so, we might call $A$ an increasing matrix or linear transformation. This question just occurred to me when I was doing generalised least square problems. I have not found any reference on this question. Could anyone help me, please? Thank you!
Update: Is there any rules to check whether $A$ has such property. In particular, does the inverse of Cholesky decomposition of a positive definite matrix have this property?