in the answer of this post: I don't understand how How $\|Ax\|\leq ||A|| ||x||$?
2026-04-07 16:32:44.1775579564
about a lemma for the proof of submultiplicativity of matrix norm
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By definition, $\|Ax\| \leq \|A\|$ for all $x$ with $\|x\|=1$. In particular, if $v$ an arbitrary vector (distinct from zero) in $\textsf K^n$ then we know that $\|tv\|=1$ where $t=1/\|v\|$.
So, $\|A(tv)\| \leq \|A\|$ and observe that $\|A(tv)\|=\|t(Av)\|=|t|\|Av\|=t\|Av\|$. In conclusion, we have $$t\|Av\|\leq\|A\|$$ and then $$\|Av\|\leq\frac1t\|A\|=\|v\|\|A\|=\|A\|\|v\|$$