Page 284 - Matrix Analysis & Applied Linear Algebra

280 Chapter 5 Norms, Inner Products, and Orthogonality

General Matrix Norms
A matrix norm is a function from the set of all complex matrices
(of all finite orders) into that satisfies the following properties.

A ≥ 0 and A =0 ⇐⇒ A = 0.
αA = |α| A for all scalars α.
(5.2.3)
A + B ≤ A + B for matrices of the same size.
AB ≤ A B for all conformable matrices.

The Frobenius norm satisfies the above definition (it was built that way),
but where do other useful matrix norms come from? In fact, every legitimate
vector norm generates (or induces) a matrix norm as described below.

Induced Matrix Norms
p
Avector norm that is defined on C for p = m, n induces a matrix
m×n
norm on C by setting
m×n n×1
A = max Ax for A ∈C , x ∈C . (5.2.4)
x =1
The footnote on p. 276 explains why this maximum value must exist.

• It’s apparent that an induced matrix norm is compatible with its
underlying vector norm in the sense that

Ax ≤ A x . (5.2.5)

1
• When A is nonsingular, min Ax = −1 . (5.2.6)
x =1 A

Proof. Verifying that max x =1 Ax satisfies the first three conditions in
(5.2.3) is straightforward, and (5.2.5) implies AB ≤ A B (see Exercise
5.2.5). Property (5.2.6) is developed in Exercise 5.2.7.
In words, an induced norm A represents the maximum extent to which

avector on the unit sphere can be stretched by A, and 1/ A −1
measures the

extent to which a nonsingular matrix A can shrink vectors on the unit sphere.
3
Figure 5.2.1 depicts this in for the induced matrix 2-norm.