Page 140 - Linear Algebra Done Right

Chapter 7. Operators on Inner-Product Spaces
128
Instead of self-adjoint, Self-Adjoint and Normal Operators
An operator T ∈L(V) is called self-adjoint if T = T . For example,
∗
2
some mathematicians if T is the operator on F whose matrix (with respect to the standard
use the term Hermitian basis) is

(in honor of the French 2 b ,
mathematician Charles 3 7
Hermite, who in 1873
then T is self-adjoint if and only if b = 3 (because M(T) =M(T ) if and
∗
published the first
only if b = 3; recall that M(T ) is the conjugate transpose of M(T)—
∗
proof that e is not the
see 6.47).
root of any polynomial
You should verify that the sum of two self-adjoint operators is self-
with integer
adjoint and that the product of a real scalar and a self-adjoint operator
coefficients).
is self-adjoint.
A good analogy to keep in mind (especially when F = C) is that
the adjoint on L(V) plays a role similar to complex conjugation on C.
A complex number z is real if and only if z = ¯ z; thus a self-adjoint
operator (T = T ) is analogous to a real number. We will see that
∗
this analogy is reflected in some important properties of self-adjoint
operators, beginning with eigenvalues.
If F = R, then by 7.1 Proposition: Every eigenvalue of a self-adjoint operator is real.
definition every
eigenvalue is real, so Proof: Suppose T is a self-adjoint operator on V. Let λ be an
this proposition is eigenvalue of T, and let v be a nonzero vector in V such that Tv = λv.
interesting only when Then
F = C.
2
λ v = λv, v
= Tv, v
= v, Tv
= v, λv
¯
2
= λ v .
¯
Thus λ = λ, which means that λ is real, as desired.
The next proposition is false for real inner-product spaces. As an
2
example, consider the operator T ∈L(R ) that is a counterclockwise
rotation of 90 around the origin; thus T(x, y) = (−y, x). Obviously
◦
2
Tv is orthogonal to v for every v ∈ R , even though T is not 0.