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.
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