# Coefficient-wise powers of matrices. Reference wanted

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 ?

• See also this post. Any such matrix $A$ also answers the question there (about the component-wise exponential). May 12, 2016 at 12:21
• It is not clear from your question whether (a) you are looking for a description, or (b) you have found a description but are unsure whether it is already known. Please clarify. May 12, 2016 at 12:28
• I indeed found a description, but wanted to know whether this is known or not. May 12, 2016 at 13:29
• @Loïc Not exactly because I want equality for $k\geqslant 1$, not $k\geqslant 0$. May 12, 2016 at 13:31
• I don't have an answer, but in searching the literature it may be useful to know that $A^{[k]}$ is called a "Hadamard power" of $A$. May 12, 2016 at 14:08