50 1. Preparatory material
1.3.5. Minors. In the previous sections, we perturbed n × n Hermitian
matrices A = An by adding a (small) n × n Hermitian correction matrix B
to them to form a new n × n Hermitian matrix A + B. Another important
way to perturb a matrix is to pass to a principal minor, for instance to the
top left n 1 × n 1 minor An−1 of An. There is an important relationship
between the eigenvalues of the two matrices:
Exercise 1.3.14 (Cauchy interlacing law). For any n×n Hermitian matrix
An with top left n 1 × n 1 minor An−1, then
(1.75) λi+1(An) λi(An−1) λi(An)
for all 1 i n. (Hint: Use the Courant-Fischer minimax theorem, The-
orem 1.3.2.) Show furthermore that the space of An for which equality
holds in one of the inequalities in (1.75) has codimension 2 (for Hermitian
matrices) or 1 (for real symmetric matrices).
Remark 1.3.8. If one takes successive minors An−1,An−2,...,A1 of an
n × n Hermitian matrix An, and computes their spectra, then (1.75) shows
that this triangular array of numbers forms a pattern known as a Gelfand-
Tsetlin pattern.
One can obtain a more precise formula for the eigenvalues of An in terms
of those for An−1:
Exercise 1.3.15 (Eigenvalue equation). Let An be an n × n Hermitian
matrix with top left n 1 × n 1 minor An−1. Suppose that λ is an
eigenvalue of An distinct from all the eigenvalues of An−1 (and thus simple,
by (1.75)). Show that
(1.76)
n−1
j=1
|uj(An−1)∗X|2
λj(An−1) λ
= ann λ
where ann is the bottom right entry of A, and X = (anj)j=1
n−1

Cn−1
is
the right column of A (minus the bottom entry). (Hint: Expand out the
eigenvalue equation Anu = λu into the
Cn−1
and C components.) Note the
similarities between (1.76) and (1.74).
Observe that the function λ
∑n−1
j=1
|uj
(An−1)∗X|2
λj (An−1)−λ
is a rational function
of λ which is increasing away from the eigenvalues of An−1, where it has a
pole (except in the rare case when the inner product
uj−1(An−1)∗X
vanishes,
in which case it can have a removable singularity). By graphing this function
one can see that the interlacing formula (1.75) can also be interpreted as a
manifestation of the intermediate value theorem.
The identity (1.76) suggests that under typical circumstances, an eigen-
value λ of An can only get close to an eigenvalue λj(An−1) if the associated
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