Suppose A and B are nonnegative matrices, that is, $A\geq 0$ and $B\geq 0$. Moreover, suppose $\|A\|_{\infty}<1, \|B\|_{\infty}<1$ and $A\leq B$. For $f(x)=x(2-x)$, we know that $f(x)$ is increasing in $(0,1)$. I wonder if it is true that $f(A)\leq f(B)$ if $f(B)\geq 0$.
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if $f$ is applied to all elements, yes. If $f(A) = A(2I-A)$ then no – Exodd Mar 16 '22 at 14:06
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Thank you for your comments – Rebecca90 Mar 17 '22 at 03:45