matrix norm

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