Page 149 - Linear Algebra Done Right

The Spectral Theorem
for the details, the most important of which is verifying that T| {u} ⊥ is
self-adjoint (this allows us to apply our induction hypothesis).
Let λ be any eigenvalue of T (because T is self-adjoint, we know 137
from the previous lemma that it has an eigenvalue) and let u ∈ V
denote a corresponding eigenvector with u = 1. Let U denote the To get an eigenvector
one-dimensional subspace of V consisting of all scalar multiples of u. of norm 1, take any
Note that a vector v ∈ V is in U ⊥ if and only if u, v = 0. nonzero eigenvector
Suppose v ∈ U . Then because T is self-adjoint, we have and divide it by its
⊥
norm.
u, Tv = Tu, v = λu, v = λ u, v = 0,
and hence Tv ∈ U . Thus Tv ∈ U ⊥ whenever v ∈ U . In other words,
⊥
⊥
U ⊥ is invariant under T. Thus we can define an operator S ∈L(U ) by
⊥
S = T| U ⊥.If v, w ∈ U , then
⊥
Sv, w = Tv, w = v, Tw = v, Sw ,
which shows that S is self-adjoint (note that in the middle equality
above we used the self-adjointness of T). Thus, by our induction hy-
pothesis, there is an orthonormal basis of U ⊥ consisting of eigenvec-
tors of S. Clearly every eigenvector of S is an eigenvector of T (because
Sv = Tv for every v ∈ U ). Thus adjoining u to an orthonormal basis
⊥
of U ⊥ consisting of eigenvectors of S gives an orthonormal basis of V
consisting of eigenvectors of T, as desired.

For T ∈L(V) self-adjoint (or, more generally, T ∈L(V) normal
when F = C), the corollary below provides the nicest possible decom-
position of V into subspaces invariant under T. On each null(T − λ j I),
the operator T is just multiplication by λ j .

7.14 Corollary: Suppose that T ∈L(V) is self-adjoint (or that F = C
and that T ∈L(V) is normal). Let λ 1 ,...,λ m denote the distinct eigen-
values of T. Then
V = null(T − λ 1 I) ⊕· · · null(T − λ m I).
Furthermore, each vector in each null(T − λ j I) is orthogonal to all vec-
tors in the other subspaces of this decomposition.

Proof: The spectral theorem (7.9 and 7.13) implies that V has a
basis consisting of eigenvectors of T. The desired decomposition of V
now follows from 5.21.
The orthogonality statement follows from 7.8.