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
 |
(6)
|
-Norm." Numer. Math. 62,
539-555, 1992.Higham, N. J. "Matrix Norms." §6.2
in