48 1. Preparatory material

Let us say that a Hermitian matrix has simple spectrum if it has no re-

peated eigenvalues. We thus see from the above exercise and (1.64) that the

set of Hermitian matrices with simple spectrum forms an open dense set in

the space of all Hermitian matrices, and similarly for real symmetric matri-

ces; thus simple spectrum is the generic behaviour of such matrices. Indeed,

the unexpectedly high codimension of the non-simple matrices (naively, one

would expect a codimension 1 set for a collision between, say, λi(A) and

λi+1(A)) suggests a repulsion phenomenon: because it is unexpectedly rare

for eigenvalues to be equal, there must be some “force” that “repels” eigen-

values of Hermitian (and to a lesser extent, real symmetric) matrices from

getting too close to each other. We now develop some machinery to make

this intuition more precise.

We first observe that when A has simple spectrum, the zeroes of the

characteristic polynomial λ → det(A − λI) are simple (i.e., the polynomial

has nonzero derivative at those zeroes). From this and the inverse function

theorem, we see that each of the eigenvalue maps A → λi(A) are smooth

on the region where A has simple spectrum. Because the eigenvectors ui(A)

are determined (up to phase) by the equations (A − λi(A)I)ui(A) = 0 and

ui(A)∗ui(A) = 1, another application of the inverse function theorem tells

us that we can

(locally15)

select the maps A → ui(A) to also be smooth.

Now suppose that A = A(t) depends smoothly on a time variable t, so

that (when A has simple spectrum) the eigenvalues λi(t) = λi(A(t)) and

eigenvectors ui(t) = ui(A(t)) also depend smoothly on t. We can then

differentiate the equations

(1.69) Aui = λiui

and

(1.70) ui

∗ui

= 1

to obtain various equations of motion for λi and ui in terms of the derivatives

of A.

Let’s see how this works. Taking first derivatives of (1.69), (1.70) using

the product rule, we obtain

(1.71)

˙

Aui + A ˙ u

i

=

˙

λ iui + λi ˙ u

i

and

(1.72) ˙ u

∗ui

i

+ ui

∗

˙ u

i

= 0.

15There may be topological obstructions to smoothly selecting these vectors globally, but

this will not concern us here as we will be performing a local analysis only. In some applications,

it is more convenient not to work with the ui(A) at all due to their phase ambiguity, and work

instead with the spectral projections Pi(A) :=

ui(A)ui(A)∗,

which do not have this ambiguity.