Matrix Norm

Prepared byDr Amin E. Khalil

Definitions:

Matrix norm is a measure of the size of a matrix, or a distance between two matrices.

p

p

Xp X

AXA

0sup

How to compute matrex norm?

The solution of matrix norm involves the following two steps:1- find candidate for the norm, say K, that satisfy the relation:

ppXKAX

2- Find at one nonzero vector that satisfy

ppXKAX 00

then KAp

Matrix L1 norm

Matrix 1-norm is the maximum of the column sum. To investigate this let m×n matrix represented in the form

yielding

Which equals

The first step is thus obtained where K= max ||Aj||p

To maximize We choose X in the form

The matrix L1 norm is the column with largest sum, thus

31A

L∞ norm

What is the infinity norm of the previous example?

The answer is 3

L2 norm

Newton-Raphson method