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I am trying to prove that $\lVert T^k x\rVert \leq \lVert T\rVert^k \lVert x\rVert$, where $T$ is an $n\times n$ matrix and $k>0$. I know that for a matrix norm induced by a vector norm this statement holds true $\lVert Ax\rVert \leq \lVert A\rVert\cdot \lVert x\rVert$. But I am not sure if it is enough to just apply that lemma to complete the proof?

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Just use induction. You already know that it is true when $n=1$. If, for a natural $n$, you have $\|T^nx\|\leqslant\|T\|^n\|x\|$, then$$\|T^{n+1}x\|=\bigl\|T^n(Tx)\bigr\|\leqslant\|T\|^n\|Tx\|\leqslant\|T\|^n\|T\|\|x\|=\|T\|^{n+1}\|x\|.$$

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Note that this follows from the more general fact that induced (matrix) norms, i.e. norms of the form $$\|T\|_{V\times V}:=\sup_{x\neq 0}\frac{\|Tx\|_V}{\|x\|_V},$$ where $\|\cdot\|_V$ is a norm on $V$, are essentially sub-multiplicative, that is: if $S,T\colon V \to V$ then $$\|ST\|_{V\times V}\leq \|S\|_{V\times V}\|T\|_{V\times V}.$$ Indeed for every $x\in V$ it holds $$\|STx\|_V=\|S(Tx)\|_V\leq \|S\|_{V\times V}\|Tx\|_V \leq \|S\|_{V\times V}\|T\|_{V\times V}\|x\|_V$$

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$$\|T^k x\| \le \|T\|\|T^{k-1} x\| \le \|T\|\|T\|\|T^{k-2} x \| \le \cdots$$

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