1.3. Eigenvalues and sums 41

The spectral theorem as stated in the introduction then follows by spe-

cialising to the case V =

Cn

and ordering the eigenvalues.

Proof. We induct on the dimension n. The claim is vacuous for n = 0, so

suppose that n ≥ 1 and that the claim has already been proven for n = 1.

Let v be a unit vector in V (thus

v∗v

= 1) that maximises the form

Re(v∗Tv);

this maximum exists by compactness. By the method of La-

grange multipliers, v is a critical point of

Re(v∗Tv)

−

λv∗v

for some λ ∈ R.

Differentiating in an arbitrary direction w ∈ V , we conclude that

Re(v∗Tw

+

w∗Tv

−

λv∗w

−

λw∗v)

= 0;

this simplifies using self-adjointness to

Re(w∗(Tv

− λv)) = 0.

Since w ∈ V was arbitrary, we conclude that Tv = λv, thus v is a unit

eigenvector of T . By self-adjointness, this implies that the orthogonal com-

plement

v⊥

:= {w ∈ V :

v∗w

= 0} of v is preserved by T . Restricting T

to this lower-dimensional subspace and applying the induction hypothesis,

we can find an orthonormal basis of eigenvectors of T on

v⊥.

Adjoining the

new unit vector v to the orthonormal basis, we obtain the claim.

Suppose we have a self-adjoint transformation A :

Cn

→

Cn,

which

of course can be identified with a Hermitian matrix. Using the orthogonal

eigenbasis provided by the spectral theorem, we can perform an orthonormal

change of variables to set that eigenbasis to be the standard basis e1,...,en,

so that the matrix of A becomes diagonal. This is very useful when dealing

with just a single matrix A; for instance, it makes the task of computing

functions of A, such as

Ak

or exp(tA), much easier. However, when one has

several Hermitian matrices in play (e.g., A, B, A + B), then it is usually not

possible to standardise all the eigenbases simultaneously (i.e., to simultane-

ously diagonalise all the matrices), except when the matrices all commute.

Nevertheless, one can still normalise one of the eigenbases to be the stan-

dard basis, and this is still useful for several applications, as we shall soon

see.

Exercise 1.3.1. Suppose that the eigenvalues λ1(A) · · · λn(A) of

an n × n Hermitian matrix are distinct. Show that the associated eigen-

basis u1(A),...,un(A) is unique up to rotating each individual eigenvec-

tor uj(A) by a complex phase

eiθj

. In particular, the spectral projections

Pj(A) :=

uj(A)∗uj(A)

are unique. What happens when there is eigenvalue

multiplicity?