Page 283 - Matrix Analysis & Applied Linear Algebra

5.2 Matrix Norms 279
5.2 MATRIX NORMS

m×n
Because C is a vector space of dimension mn, magnitudes of matrices
m×n mn
A ∈C can be “measured” by employing any vector norm on C . For

2 −1
example, by stringing out the entries of A = into a four-component
−4 −2
4
vector, the euclidean norm on can be applied to write
1/2
2 2 2 2
A = 2 +(−1) +(−4) +(−2) =5.
This is one of the simplest notions of a matrix norm, and it is called the Frobenius
(p. 662) norm (older texts refer to it as the Hilbert–Schmidt norm or the Schur
norm). There are several useful ways to describe the Frobenius matrix norm.
Frobenius Matrix Norm
m×n
The Frobenius norm of A ∈C is defined by the equations

2
2
2
2
∗
A = |a ij | = A i∗ = A ∗j = trace (A A). (5.2.1)
F 2 2
i,j i j
The Frobenius matrix norm is fine for some problems, but it is not well suited
for all applications. So, similar to the situation for vector norms, alternatives need
to be explored. But before trying to develop different recipes for matrix norms, it
makes sense to first formulate a general definition of a matrix norm. The goal is
to start with the defining properties for a vector norm given in (5.1.9) on p. 275
and ask what, if anything, needs to be added to that list.
Matrix multiplication distinguishes matrix spaces from more general vector
spaces, but the three vector-norm properties (5.1.9) say nothing about products.
So, an extra property that relates AB to A and B is needed. The
Frobenius norm suggests the nature of this extra property. The CBS inequality
2 2 2 2 2 2
insures that Ax = |A i∗ x| ≤ A i∗ x = A x . That is,
2 i i 2 2 F 2
Ax 2 ≤ A x , (5.2.2)
F 2
and we express this by saying that the Frobenius matrix norm
and the
F
euclidean vector norm
are compatible. The compatibility condition (5.2.2)
2
implies that for all conformable matrices A and B,
2 2 2 2 2
AB = [AB] ∗j = AB ∗j ≤ A B ∗j
F 2 2 F 2
j j j
2 2 2 2
= A B ∗j = A B =⇒ AB ≤ A B .
F 2 F F F F F
j
This suggests that the submultiplicative property AB ≤ A B should be
added to (5.1.9) to define a general matrix norm.