Frobernius Norm

Description:

• Very similar to Euclidean Length applied to the vector formed by all elements of $A\in {\mathbb{R}}^{m.n}$
  • $||A|{|}_F=\sqrt{\text{trace }A{A}^{⊺}}=\sqrt{\sum _{i=1}^m\sum _{j=1}^n|A(i,j){|}^2}$
• For any $x\in {\mathbb{R}}^n$, it holds that $||Ax|{|}_2\le ||A|{|}_F||x|{|}_2$
  • consequence of Cauchy-Schwartz Inequality applied to $|a_i^{⊺}x|$, where $a_i$ is a generic row of $A$
• The Frobenius norm is sub-multiplicative:
  • for any $B\in R^{n,p}$ , it holds that $||AB|{|}_F\le ||A|{|}_F||B|{|}_F$

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