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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?2012-08-11
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    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?2012-08-11
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    @neelp: in general $B$ is neither guaranteed to exist nor be unique.2012-08-11
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    That is true. The question must have meant for $r$ to be a positive integer. Right?2012-08-11
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    Then the simplest recursion one can think of seems to yield the proof, doesn't it?2012-08-11
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    Yes. Then the problem is easy.2012-08-11
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    @neelp You could write an answer.2012-08-13

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