Matrix Norm
Given a square complex or real matrix 
, a matrix norm 
is a nonnegative number associated with 
having the properties
1. 
when 
and 
iff 
,
2. 
for any scalar 
,
3. 
,
4. 
.
Let 
, ..., 
be the eigenvalues of 
, then
 |
(1) |
The matrix 
-norm is defined for a real number 
and a matrix 
by
 |
(2) |
where 
is a vector norm. The task of computing a matrix 
-norm is difficult for 
since it is a nonlinear optimization problem with constraints.
Matrix norms are implemented as Norm[m, p], where 
may be 1, 2, Infinity, or "Frobenius".
The maximum absolute column sum norm 
is defined as
 |
(3) |
The spectral norm 
, which is the square root of the maximum eigenvalue of 
(where 
is the conjugate transpose),
 |
(4) |
is often referred to as "the" matrix norm.
The maximum absolute row sum norm is defined by
 |
(5) |
, 
, and 
satisfy the inequality
 |