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Much work done on computers with vectors and matrices is approximation mathematics, and it is necessary to be able to say when one vector is near another, or when a vector is small, and similarly for matrices. For this purpose the idea of norm is introduced. In most cases, the norm of a 1-dimensional vector or matrix is the absolute value of the number. We begin with three vector norms in common use:
(i) Euclidean norm:
(ii) Maximum, Chebysher or sup norm:
(iii) 1-norm:
These three norms, which are non-negative functions on the
n-dimensional vector space or , satisfy the following
vector norm axioms:
(V1)
and if and only if x = 0
(V2)
(Homogeneity)
(V3)
(Triangle
Inequality)
A Consequence of (V3) is
(Reverse Triangle
Inequality)
We now discuss some matrix norms:
i) Schur or Frobenius norm:
ii) Max absolute row sum norm:
iii) Max absolute column sum norm:
iv) 2-norm: = [dominant eigen value of
It can be verified that these satisfy the matrix norm axioms
:
(M1)
if and only if A =0
(M2)
a scalar.
(M3)
(M4)
Example: Compute the 1,-2-,-, and Frobenious norms of the
matrix