4.9b Operator Norm

Operator Norm

The ${\ell }^p$-induced operator norm of a matrix $A\in {\mathbb{R}}^{m\times n}$, given a vector p-norm, is the maximum normalized amount by which $A$ can stretch any nonzero vector, such that
$\Vert A{\Vert }_p=\underset{\vec{x}\ne 0}{\max }\frac{\Vert A\vec{x}{\Vert }_p}{\Vert \vec{x}{\Vert }_p}=\underset{\Vert \vec{x}{\Vert }_p\le 1}{\max }\Vert A\vec{x}{\Vert }_p,$
where the latter comes from the fact that the operator norm measures the magnitude relative to the unit norm (it is normalized). Then:

• The ${\ell }^1$-induced norm is the largest absolute column sum, $\Vert A{\Vert }_1=\underset{j=1,\dots ,n}{\max }\sum _{i=1}^m|A_{ij}|$.
• The ${\ell }^{\infty }$-induced norm is the largest absolute row sum, $\Vert A{\Vert }_{\infty }=\underset{i=1,\dots ,m}{\max }\sum _{j=1}^n|A_{ij}|$.
• The ${\ell }^2$-induced (spectral) norm corresponds to the square root of the largest eigenvalue of $A^{T}A$, such that $\Vert A{\Vert }_2=\sqrt{\underset{i=1,\dots ,n}{\max }{\lambda }_i}$.

Properties

(Submultiplication) $\Vert AB{\Vert }_p\le \Vert A{\Vert }_p\Vert B{\Vert }_p$