Let $A=[[a_{ij}]]$ and $B=[[b_{ij}]]$ be two positive semi-definite matrices of same dimensions. Further they have a property that, if $a_{ij}=0$ then $b_{ij}=0$ (i.e. the nonzero entries appear in the same positions). Construct a matrix $M=[[m_{ij}]]$, where $m_{ij}=\max\{a_{ij},b_{ij}\}$. The question is whether $M$ is positive semidefinite?
My could not give a proof or construct a counter example. I, can also restrict the case, when $A$ and $B$ are of unit trace. Advanced thanks for any help.
It's easy to construct a counterexample. Consider, e.g. $$ A=\pmatrix{1&1-\varepsilon&0\\ 1-\varepsilon&1&1-\varepsilon\\ 0&1-\varepsilon&1} =\text{entrywise maximum of } B \text{ and } C, $$ where $$ B=\pmatrix{1&\varepsilon&0\\ \varepsilon&1&1-\varepsilon\\ 0&1-\varepsilon&1}, \ C=\pmatrix{1&1-\varepsilon&0\\ 1-\varepsilon&1&\varepsilon\\ 0&\varepsilon&1}. $$ When $\varepsilon>0$ is small, both $B$ and $C$ are positive definite, but $A$ is indefinite.