Both $A$ and $B$ are finite-dimensional matrices. Why is $\|AB\| \leq \|A\| \, \|B\|$?
2026-03-27 13:20:12.1774617612
Why is $\|AB\| \leq \|A\| \, \|B\|$?
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$\|A\|=\sup_{\|x\|=1}\|A(x)\|$. This implies that for $x\neq 0$, $\|A({x\over{\|x\|}})\leq \|A\|$. We deduce that $\|A(x)\|\leq\|A\|\|x\|$.
This implies that $\|AB\|=Sup_{\|x\|=1}\|A(B(x))\|\leq sup_{\|x\|=1}\|A\|\|B(x)\|\leq sup_{\|x\|=1}\|A\|\|B\|\|x\|=\|A\|\|B\|$.