Page 150 - Linear Algebra Done Right

Chapter 7. Operators on Inner-Product Spaces
138
Normal Operators on Real
Inner-Product Spaces
The complex spectral theorem (7.9) gives a complete description
of normal operators on complex inner-product spaces. In this section
we will give a complete description of normal operators on real inner-
product spaces. Along the way, we will encounter a proposition (7.18)
and a technique (block diagonal matrices) that are useful for both real
and complex inner-product spaces.
We begin with a description of the operators on a two-dimensional
real inner-product space that are normal but not self-adjoint.

7.15 Lemma: Suppose V is a two-dimensional real inner-product
space and T ∈L(V). Then the following are equivalent:

(a) T is normal but not self-adjoint;
(b) the matrix of T with respect to every orthonormal basis of V
has the form
a −b
b a ,
with b = 0;
(c) the matrix of T with respect to some orthonormal basis of V has
the form

a −b
,
b a
with b> 0.

Proof: First suppose that (a) holds, so that T is normal but not
self-adjoint. Let (e 1 ,e 2 ) be an orthonormal basis of V. Suppose

a c
7.16 M T, (e 1 ,e 2 ) = .
b d
2
2
2
2
2
2
Then Te 1 = a + b and T e 1 = a + c . Because T is normal,
∗
2
2
Te 1 = T e 1 (see 7.6); thus these equations imply that b = c .
∗
Thus c = b or c =−b. But c = b because otherwise T would be self-
adjoint, as can be seen from the matrix in 7.16. Hence c =−b,so

a −b

7.17 M T, (e 1 ,e 2 ) = .
b d