Matrix Norm

Description:

• A function $f:R^{m,n}\to R$ is a matrix norm if, analogously to the vector norm case, it satisfies three standard axioms. Namely, for all $A,B\in R^{m,n}$ and all $\alpha \in R$:
  • $f(A)\ge 0$. Furthermore $f(A)=0$ if and only if $A=0$;
  • $f(\alpha A)=|\alpha |f(A)$
  • $f(A+B)\le f(A)+f(B)$
• Many of the popular matrix norms also satisfy a fourth condition called sub-multiplicativity:
  • for any conformably sized matrices $A,B:f(AB)\le f(A)f(B)$

Frobernius Norm

Operator Norm