Matrices of the form $A^p=(a_{ij}^p)$

Question

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

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.

For the first bullet point, there is always such a $p$, namely $p=1$. — Marc van Leeuwen, Apr 10 '14 at 07:15
Diagonal matrices have property (1). If $n=1$ these identities hold trivially ;) — daw, Apr 10 '14 at 18:44
I think the OP is really asking "for which matrices $A$ except the diagonal ones, do we have $A^p=(a_{ij}^p)$ ?" — Ewan Delanoy, Oct 02 '14 at 09:41

score 1 · Answer 1 · edited Apr 13 '17 at 12:20

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

Matrices of the form $A^p=(a_{ij}^p)$

1 Answers1