# Entries of a matrix raised to any real power

Suppose $A=(a_{ij})$ is an $n\times n$ real matrix and define $T(A)=\max\{|a_{ij}|\}$, where the maximum is taken over $1\leq i,j \leq n$.

I know how to show that $T(AB)\leq nT(A)T(B)$ for all $A$ and $B$.

Show that $T(A^{r})\leq n^{r-1}(T(A))^{r}$ for all $A$ and all $r\geq 1, r\in\mathbb{R}$.

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How do you define real power of a matrix? – Davide Giraudo Aug 11 '12 at 21:01
I am also not sure about that. I was just assuming that for a rational number, $A^{p/q}$ is some matrix $B$ such that $B^{q}=A^{p}$. And for a real number, $r$, $A^{r}$ can be a limit of $A$ raised to the power of rational numbers approaching $r$. Any thoughts? – neelp Aug 11 '12 at 21:08
@neelp: in general $B$ is neither guaranteed to exist nor be unique. – Qiaochu Yuan Aug 11 '12 at 21:08
Then the simplest recursion one can think of seems to yield the proof, doesn't it? – Did Aug 11 '12 at 21:58
@neelp You could write an answer. – Davide Giraudo Aug 13 '12 at 14:47