Matrix Norms

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

• 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.
• 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 $\vert\vert A\vert\vert _\infty$.
• 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.