Page 76 - Textos de Matemática Vol. 44

66 M. E. HOCHSTENBACH, D. A. SINGER AND P. F. ZACHLIN
Ernst [1] note that
−1 x∗A−1x y∗A∗y y∗y ν(A) (4.1) ν(A )=min =min · ≥ .
x̸=0 x∗x y̸=0 y∗y y∗A∗Ay ∥A∥2
Using similar techniques we now derive new upper bounds for ν(A−1), and
upper and lower bounds for μ(A−1).
ν(A) −1
ν(A) μ(A)
) ≤ min σ2 min
min min
Proof. With an eye on (4.1) we have
Proposition 4.1. (a) ∥A∥2 ≤ ν(A ν(A) μ(A) −1
(A) , ∥A∥2 .
μ(A) (b)max σ2 (A),∥A∥2 ≤μ(A )≤σ2 (A).
−1 y∗A∗y y∗y ν(A )≤min ·max =
y̸=0 y∗y y̸=0 y∗A∗Ay ∥A∥2
ν(A) y̸=0 y∗y y̸=0 y∗A∗Ay σ2 (A)
.
Also,
Part (b) follows from
−1 y∗A∗y y∗y μ(A) ν(A )≤max ·min =
.
min
−1 x∗A−1x y∗A∗y y∗y μ(A )=max ≤max ·max =
μ(A) x̸=0 x∗x y̸=0 y∗y y̸=0 y∗A∗Ay σ2 (A)
,
−1 y∗A∗y y∗y ν(A) μ(A )≥min ·max =
,
min
and
−1 y∗A∗y y∗y μ(A) μ(A )≥max ·min = .
y̸=0 y∗y y̸=0 y∗A∗Ay σ2 (A) min
y̸=0 y∗y y̸=0 y∗A∗Ay ∥A∥2 In particular, we have from this

∥A∥ ≤ (ν(A−1))−1 μ(A) ≤ (ν(A−1))−1 ∥A∥
and therefore ∥A∥ ≤ min{2 μ(A), ν(A−1)−1}.
From a different perspective the bounds in this subsection, together with
bounds such as μ(A−1) ≤ ∥A−1∥ ≤ 2μ(A−1), imply bounds for ∥A∥, ∥A−1∥ and the condition number κ(A) = ∥A∥ ∥A−1∥ in terms of W (A) and W (A−1).