Page 286 - Matrix Analysis & Applied Linear Algebra

282 Chapter 5 Norms, Inner Products, and Orthogonality

using the method of Lagrange multipliers. Introduce a new variable λ (the
Lagrange multiplier), and consider the function h(x,λ)= f(x) − λg(x). The
points at which f is maximized are contained in the set of solutions to the
equations ∂h/∂x i =0 (i =1, 2,...,n) along with g(x)=1. Differentiating
h with respect to the x i ’s is essentially the same as described on p. 227, and
T
the system generated by ∂h/∂x i =0 (i =1, 2,...,n)is(A A − λI)x = 0. In
T
other words, f is maximized at a vector x for which (A A − λI)x = 0 and
T
x =1. Consequently, λ must be a number such that A A − λI is singular
2
(because x = 0 ). Since
T
T
T
x A Ax = λx x = λ,
it follows that
1/2

T
T
A = max Ax = max Ax = max x A Ax = λ max ,
2
x =1 x 2 =1 x T x=1
T
where λ max is the largest number λ for which A A − λI is singular. A similar
argument applied to (5.2.6) proves (5.2.8). Also, an independent development of
(5.2.7) and (5.2.8) is contained in the discussion of singular values on p. 412.
Example 5.2.1
Problem: Determine the induced norm A 2 as well as A −1 2 for the non-
singular matrix

1 3 −1
A = √ √ .
3 0 8
T
Solution: Find the values of λ that make A A − λI singular by applying
Gaussian elimination to produce

3 − λ −1 −1 3 − λ −1 3 − λ
T
A A − λI = −→ −→ 2 .
−1 3 − λ 3 − λ −1 0 −1+(3 − λ)
T
2
This shows that A A−λI is singular when −1+(3−λ) =0 or, equivalently,
when λ =2 or λ =4, so λ min =2 and λ max =4. Consequently, (5.2.7) and
(5.2.8) say that
−1 1 1
A 2 = λ max =2 and A 2 = √ = √ .
2
λ min
T
Note: As mentioned earlier, the values of λ that make A A − λI singular
T
are called the eigenvalues of A A, and they are the focus of Chapter 7 where
their determination is discussed in more detail. Using Gaussian elimination to
determine the eigenvalues is not practical for larger matrices.

Some useful properties of the matrix 2-norm are stated below.