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: $A^{[k]} =(a\_{ij}^k )\_{1\leqslant i,j\leqslant n}$. 



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> 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 ?
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Coefficient-wise powers of matrices. Reference wanted
This is problem 50.4 by Moubinool Omarjee from [volume 50](http://www.ilasic.org/IMAGE/IMAGES/image50.pdf) of The Bulletin of the International Linear Algebra Society, with solutions in [volume 51](http://www.ilasic.org/IMAGE/IMAGES/image51.pdf) by Eugene Herman and Bojan Kuzma, and further work by Roman Drnovsek in [When powers of a matrix coincide with its Hadamard powers.](https://arxiv.org/abs/1509.00241)


For real matrices the solution is


![](https://ilorentz.org/beenakker/MO/Hadamard_product.png)


