Linear Algebra, Theory And Applications, 2012a

404 POSITIVE MATRICES
Proof: Let λ ∈ S. Then there exists x >> 0 such that Ax >λx. Consider y ≡ x/ ||x|| 1
.
Then ||y|| 1
= 1 and Ay >λy. Therefore, λ ∈ S 1 and so S ⊆ S 1 . Therefore, sup (S) ≤
sup (S 1 ) .
Now let λ ∈ S 1 . Then there exists x ≥ 0 such that ||x|| 1
=1sox > 0 and Ax >λx.
Letting y ≡ Ax, it follows from Lemma A.0.2 that Ay >> λy and y >> 0. Thus λ ∈ S
and so S 1 ⊆ S which shows that sup (S 1 ) ≤ sup (S) .
This lemma is significant because the set, {x ≥ 0 such that ||x|| 1
=1}≡K is a compact
set in R n . Define
λ 0 ≡ sup (S) =sup(S 1 ) . (1.1)
The following theorem is due to Perron.
Theorem A.0.4 Let A>>0 be an n × n matrix and let λ 0 be given in (1.1). Then
1. λ 0 > 0 and there exists x 0 >> 0 such that Ax 0 = λ 0 x 0 so λ 0 is an eigenvalue for A.
2. If Ax = μx where x ≠ 0, and μ ≠ λ 0 . Then |μ| 0, consider the vector, e ≡ (1, ··· , 1) T .Then
(Ae) i
= ∑ j
A ij > 0
and so λ 0 is at least as large as
min
i
∑
A ij .
j
Let {λ k } be an increasing sequence of numbers from S 1 converging to λ 0 . Letting x k be
the vector from K which occurs in the definition of S 1 , these vectors are in a compact set.
Therefore, there exists a subsequence, still denoted by x k such that x k → x 0 ∈ K and
λ k → λ 0 . Then passing to the limit,
Ax 0 ≥ λ 0 x 0 , x 0 > 0.
If Ax 0 >λ 0 x 0 , then letting y ≡ Ax 0 , it follows from Lemma A.0.2 that Ay >> λ 0 y and
y >> 0. But this contradicts the definition of λ 0 as the supremum of the elements of S
because since Ay >> λ 0 y, it follows Ay >> (λ 0 + ε) y for ε a small positive number.
Therefore, Ax 0 = λ 0 x 0 . It remains to verify that x 0 >> 0. But this follows immediately
from
0 < ∑ A ij x 0j =(Ax 0 ) i
= λ 0 x 0i .
j
This proves 1.
Next suppose Ax = μx and x ≠ 0 and μ ≠ λ 0 . Then |Ax| = |μ||x| . But this implies
A |x| ≥|μ||x| . (See the above abominable definition of |x|.)
Case 1: |x| ̸= x and |x| ≠ −x.
In this case, A |x| > |Ax| = |μ||x| and letting y = A |x| , it follows y >> 0 and
Ay >> |μ| y which shows Ay >> (|μ| + ε) y for sufficiently small positive ε and verifies
|μ|