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?
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