Induced matrix norm
= Induced matrix norm
{title2=$\|B\|=\sup_{x\ne0}\|Bx\|/\|x\|$}
A <vector norm> on a finite-dimensional real or complex <vector space> induces the displayed <matrix norm> on its <linear operators>. It is the corresponding <operator norm>, and the <supremum> can equivalently be taken over the <unit sphere>. Applying the length bound twice proves <submultiplicativity>: $\|BCx\|\leq\|B\|\|C\|\|x\|$, hence $\|BC\|\leq\|B\|\|C\|$.