Matrices of the form $A^p=(a_{ij}^p)$ I am wondering if there is a name for these kind of matrices and if they are interesting or not? Do they even exist?
Let $A$ be a $n\times n$ matrix with elements $a_{ij}$. $A= (a_{ij})_{i,j\in\{1, \dotsc, n\}}$.


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*For some integer (maybe rational) $p$, $A^p= (a_{ij}^p)_{i,j\in\{1, \dotsc, n\}}$

*$\exp A= (\exp a_{ij})_{i,j\in\{1, \dotsc, n\}}$


Do these matrices exist? Do they have names? If diagonal matrices are like this, What about others?
Thank you for your time, help.

I quite don't understand your question. Of course you can define $A^p$, if $\forall i,j\in\{1,\dots,n\}:(a_{ij})^p$ makes sense. (As matrix is defined as schema of elements, not necessarily (real) numbers). Also, it does not require the matrix to be square.
However, it don't think they can be used for something.
However, there is a way you can calculate for example $sin(A)$, where $A$ is square matrix. More reading can be found for example here $\sin(A)$, where $A$ is a matrix
