Page 146 - Linear Algebra Done Right

Chapter 7. Operators on Inner-Product Spaces
134
(because a 1,2 = 0, as we showed in the paragraph above) and
2
2
2
2
T e 2 =|a 2,2 | +|a 2,3 | + ··· + |a 2,n | .
∗
Because T is normal, Te 2 = T e 2 . Thus the two equations above
∗
imply that all entries in the second row of the matrix in 7.10, except
possibly the diagonal entry a 2,2 , equal 0.
Continuing in this fashion, we see that all the nondiagonal entries
in the matrix 7.10 equal 0, as desired.
We will need two lemmas for our proof of the real spectral theo-
rem. You could guess that the next lemma is true and even discover its
proof by thinking about quadratic polynomials with real coefficients.
2
Specifically, suppose α, β ∈ R and α < 4β. Let x be a real number.
This technique of Then
completing the square 2

2
can be used to derive x + αx + β = x + α 2 + β − α
the quadratic formula. 2 4
> 0.
2
In particular, x + αx + β is an invertible real number (a convoluted
way of saying that it is not 0). Replacing the real number x with a
self-adjoint operator (recall the analogy between real numbers and self-
adjoint operators), we are led to the lemma below.

7.11 Lemma: Suppose T ∈L(V) is self-adjoint. If α, β ∈ R are such
2
that α < 4β, then
2
T + αT + βI
is invertible.

2
Proof: Suppose α, β ∈ R are such that α < 4β. Let v be a nonzero
vector in V. Then

2 2
(T + αT + βI)v, v = T v, v + α Tv, v + β v, v
2
= Tv, Tv + α Tv, v + β v
2 2
≥ Tv −|α| Tv v + β v
2
|α| v 2 α 2
= Tv − + β − v
2 4
> 0,