Operator Norm

Description:

• Give a characterization of the maximum input-output gain of the linear map $u\to y=Au$.
• Choosing to measure both inputs and outputs in terms of a given lp norm
  • with typical values $p=1,2,...,\infty$, leads to the definition
  • $||A|{|}_p\doteq {\max }_{u\ne 0}\frac{||Au|{|}_p}{||u|{|}_p}={\max }_{||u||=1}||Au|{|}_p$
• By definition, for every $u$,$||Au|{|}_p\le ||A|{|}_p||u|{|}_p$ .
  • From this property follows that any operator norm is sub-multiplicative, that is, for any two conformably sized matrices A, B, it holds that $||AB|{|}_p\le ||A|{|}_p||B|{|}_p$
• This fact is easily seen by considering the product $AB$ as the series connection of the two operators $B,A$:
  • $||Bu|{|}_p\le ||B|{|}_p||u|{|}_p,|ABu|p\le ||A|{|}_p||Bu|{|}_p\le ||A|{|}_p||B|{|}_p||u|{|}_p$

$l_1$-induced norm:

• Corresponds to the largest absolute column sum:
  • $||A|{|}_1=\underset{||u|{|}_1=1}{\max }||Au|{|}_1=\underset{j=1,...,n}{\max }\sum _{i=1}^m|a_{ij}|$
  • For each column, get the sum of absolute and find the max

$l_2$-induced norm:

• Spectral norm
• Corresponds to the square-root of the largest singular value of $A$
  • $||A|{|}_2=\underset{||u|{|}_2=1}{\max }||Au|{|}_2={\sigma }_{\max }(A)$

$l_{\infty }$-induced norm:

• Corresponds to the largest absolute row sum:
  • $||A|{|}_1\infty =\underset{||u|{|}_{\infty }=1}{\max }||Au|{|}_{\infty }=\underset{j=1,...,m}{\max }\sum _{j=1}^n|a_{i}j|$
  • For each row, get the sum of absolute and find the max