TITLE: Coefficient-wise powers of matrices. Reference wanted
QUESTION [10 upvotes]: Let $K$ be a commutative field and ${\rm M}_n (K)$ be the ring of $n\times n$ square matrices with coefficients in $K$ ($n\geqslant 1$ is an integer). For $k\geqslant 1$ and $A =(a_{ij})_{1\leqslant i,j\leqslant n}\in {\rm M}_n (K)$, define: $A^{[k]} =(a_{ij}^k )_{1\leqslant i,j\leqslant n}$. 

Is the description of all matrices $A\in {\rm M}_n (K)$ satisfying $A^k =A^{[k]}$,  for all $k\geqslant 1$,  known? If yes do you have a reference ?

REPLY [17 votes]: This is problem 50.4 by Moubinool Omarjee from volume 50 of The Bulletin of the International Linear Algebra Society, with solutions in volume 51 by Eugene Herman and Bojan Kuzma, and further work by Roman Drnovsek in When powers of a matrix coincide with its Hadamard powers.
For real matrices the solution is