Norms

• Norm, a measure of the magnitude of the matrix.
• The magnitude of a single number is just its distance from zero:
  $\vert-4.2\vert = 4.2$.
• For vectors in two- or three space, norm is called the Euclidean norm, and is computed by $\sqrt{x_1^2+x_2^2+x_3^2}$.
• We compute the Euclidean norm of vectors with more than three components by
  
  $$\vert\vert x\vert\vert _e=\sqrt{x_1^2+x_2^2+\ldots+x_n^2}=\left( \sum_{i=1}^n x_i^2 \right)^{1/2}$$
  
  

• Defining the $p$-norm as
  
  $$\vert\vert x\vert\vert _p=\left( \sum_{i=1}^n \vert x_i\vert^p \right)^{1/p}$$
  
  

• Using $\vert\vert A\vert\vert$ to represent the norm of matrix A, some properties
  1. $\vert\vert A\vert\vert \geq 0$ and $\vert\vert A\vert\vert=0$ if and only if $A=0$
  2. $\vert\vert kA\vert\vert=\vert\vert k\vert\vert\vert\vert A\vert\vert$
  3. $\vert\vert A+B\vert\vert \leq \vert\vert A\vert\vert+\vert\vert B\vert\vert$
  4. $\vert\vert AB\vert\vert \leq \vert\vert A\vert\vert\vert\vert B\vert\vert$
• 1-, 2-, and $\infty$-norms;
  
  $$\begin{array}{l} \vert\vert x\vert\vert _1=\sum_{i=1}^n \ve... ...i \leq n} \vert x_i\vert =maximum-magnitude~norm\\ \end{array}$$
  
  

• i.e., Compute the 1-, 2-, and $\infty$-norms of the vector $x=(1.25, 0.02, -5.15, 0)$
  
  $$\begin{array}{l} \vert\vert x\vert\vert _1=\vert 1.25\vert+... ....2996\\ \vert\vert x\vert\vert _\infty = 5.15 \\ \end{array}$$