2. Norms of Vectors and Matrices

Much work done on computers with vectors and matrices is approximation mathematics, and it is necessary to be able to say when one vector is near another, or when a vector is small, and similarly for matrices. For this purpose the idem of norm is introduced. In most cases, the norm of a 1-dimensional vector or matrix is the absolute value of the number. We begin with three vector norms in common use:
(i) Euclidean norm: $\vert\vert x\vert\vert _2=[\sum\vert x_i\vert^2]^{1/2}]$
(ii) Maximum,Chebysher or sup norm: $\vert\vert x\vert\vert _{\infty}=max\vert x_i\vert$
(iii) 1-norm: $\vert\vert x\vert\vert _1=\sum \vert x_i\vert$
These three norms, which are non-negative functions on the n-dimensional vector space $R_n$ or $C_n$, satisfy the following vector norm axioms:
(V1) $\vert\vert x\vert\vert \geq 0$ and $\vert\vert x\vert\vert=0$ if and only if x = 0
(V2) $\vert\vert\alpha x\vert\vert = \vert\alpha\vert \,\vert\vert x\vert\vert$                     (Homogeneity)
(V3) $\vert\vert x+y\vert\vert \leq \vert\vert x\vert\vert + \vert\vert y\vert\vert$          (Triangle Inequality)

A Consequence of (V3) is
$\vert\vert x-y\vert\vert \geq \left\vert \vert\vert x\vert\vert-\vert\vert y\vert\vert\right\vert$                     (Reverse Triangle Inequality)
We now discuss some matrix norms:
i) Schur or Frobenius norm: $\vert\vert A\vert\vert _F=[\sum\sum\vert a_{ij}\vert^2]^{1/2}$
ii) Max absolute row sum norm: $\vert\vert A\vert\vert _{\infty}=$max$\sum\limits_{j}\vert a_{ij}\vert$
iii) Max absolute column sum norm: $\vert\vert A\vert\vert _i=$max$\sum\limits_i\vert a_{ij}\vert$
iv) 2-norm: $\vert\vert A\vert\vert _2$ = [dominant eigen value of AA]$^{1/2}$
It can be verified that if these satisfy the matrix norm axioms :
(M1) $\vert\vert A\vert\vert \geq 0 \,\, , \vert\vert A\vert\vert =0$ if and only if A =0
(M2) $\vert\vert\alpha A\vert\vert =\vert\alpha\vert \vert\vert A\vert\vert \qquad \alpha,$ a scalar.
(M3) $\vert\vert A+B\vert\vert \leq \vert\vert A\vert\vert + \vert\vert B\vert\vert$
(M4) $\vert\vert AB\vert\vert \leq \vert\vert A\vert\vert \,\vert\vert B\vert\vert$
Example: Compute the 1,-2-,$\infty$-, and Frobenious norms of the matrix

$$A=\left[% \begin{array}{ccc} 5 & -5 & -7 \\ -4 & 2 & -4 \\ -7 & -4 & 5 \\ \end{array}% \right]$$

$$\vert\vert A\vert\vert _F =15\,\,; \,\, \vert\vert A\vert\vert _{... ...\,\,;\,\,\vert\vert A\vert\vert _1=16\,\, ; \,\,\vert\vert A\vert\vert _2=12.03$$