Matrix Norms

• The norms of a matrix are similar to the norms of a vector.
  
  $$\begin{array}{l} \vert\vert A\vert\vert _1= max_{1\leq j \le... ...} \sum_{j=1}^n \vert a_{ij}\vert=maximum~row~sum\\ \end{array}$$
  
  

• Suppose $r$ is the largest eigenvalue of $A^T*A$. Then $\vert\vert A\vert\vert _2=r^{1/2}$. This is called the spectral norm of $A$, and $\vert\vert A\vert\vert _2$ is always less than (or equal to) $\vert\vert A\vert\vert _1$ and $\vert\vert A\vert\vert _\infty$.
• For an $m \times n$ matrix, the Frobenius norm is defined as
  
  $$\vert\vert A\vert\vert _f =\left(\sum_{i=1}^m \sum_{j=1}^n a_{ij}^2 \right)^{1/2}$$
  
  
  The Frobenius norm is a good measure of the magnitude of a matrix.
• The spectral norm is usually the most expensive. Which norm is best? In most instances, we want the norm that puts the smallest upper bound on the magnitude of the matrix. In this sense, the spectral norm is usually the ''best''.

  A =
  	5 -5 -7
   -4  2 -4
   -7 -4  5
  >> norm(A,'fro')
  ans =
  15
  >> norm(A,inf)
  ans =
  17
  >> norm(A,1)
  ans=
  16
  >> norm(A)
  ans =
  12.0301
  >> norm (A,2)
  ans =  
  12.0301
  

  we observe that the 2-norm, the spectral norm, is the norm we get if we just ask for the norm. The smallest norm of the matrix is the spectral norm, it is the tightest measure.