Page 137 - Textos de Matemática Vol. 44

Proof. The claim is equivalent to
∥An∥2 <(n−1)2a2 +(n−1)a,
and further, since
equivalent to
∥An∥2 = (4n−6)a2 +n2 −5n+6,
LARGEST EIGENVALUE 127
(a2 −1)n2 −(6a2 −a−5)n+7a2 −a−6>0, which holds for large n.
Proposition 4.2. If A = An and n is large enough, then u < uOS. Proof. Sincer1 =(n−1)aandm=1,wehave
uOS = (n−1)a− 1 n−2 + 1 n−1ρ, (n−1)a (n−1)a
where
Therefore
On the other hand,
ρ= 1 2(2n−3)a+(n−2)(n−3) . n
lim uOS n→∞ (n − 1)a
= 1.
u2
n→∞ (n−1) a n→∞ (n−1) a a
∥An∥2 1 2 2 = 2,
5. Examples
We saw already that u can beat uOS and uBG if n is large enough. Now we
study examples where n is small. Example 5.1. Let
0 1 2
A=1 0 5, λ=5.784.
250 EvenuF =7beatsu=7.262.
lim
2 2 = lim
lim u = 1 < 1,
so
and the claim follows.
n→∞ (n − 1)a a