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