Page 285 - Matrix Analysis & Applied Linear Algebra

5.2 Matrix Norms 281

1

max Ax = A
x =1

A
1
min Ax =
-1
x =1 A
3
Figure 5.2.1. The induced matrix 2-norm in .
Intuition might suggest that the euclidean vector norm should induce the
Frobenius matrix norm (5.2.1), but something surprising happens instead.

Matrix 2-Norm

• The matrix norm induced bythe euclidean vector norm is

A = max Ax = λ max , (5.2.7)
2
2
x =1
2
where λ max is the largest number λ such that A A − λI is singular.
∗
• When A is nonsingular,
1 1

A −1 = = √ , (5.2.8)
2
min Ax 2 λ min
x =1
2
where λ min is the smallest number λ such that A A−λI is singular.
∗
Note: If you are alreadyfamiliar with eigenvalues, these saythat λ max
and λ min are the largest and smallest eigenvalues of A A (Example
∗
7.5.1, p. 549), while (λ max ) 1/2 = σ 1 and (λ min ) 1/2 = σ n are the largest
and smallest singular values of A (p. 414).

Proof. To prove (5.2.7), assume that A m×n is real (a proof for complex ma-
2
trices is given in Example 7.5.1 on p. 549). The strategyis to evaluate A by
2
solving the problem
2 T T T
maximize f(x)= Ax = x A Ax subject to g(x)= x x =1
2