40 1. Preparatory material

(1.52) and (1.54) are

examples12.

These eigenvalue inequalities can mostly

be deduced from a number of minimax characterisations of eigenvalues (of

which (1.53) is a typical example), together with some basic facts about

intersections of subspaces. Examples include the Weyl inequalities

(1.55) λi+j−1(A + B) ≤ λi(A) + λj(B),

valid whenever i, j ≥ 1 and i + j − 1 ≤ n, and the Ky Fan inequality

λ1(A + B) + · · · + λk(A + B) ≤

λ1(A) + · · · + λk(A) + λ1(B) + · · · + λk(B).

(1.56)

One consequence of these inequalities is that the spectrum of a Hermitian

matrix is stable with respect to small perturbations.

We will also establish some closely related inequalities concerning the

relationships between the eigenvalues of a matrix, and the eigenvalues of its

minors.

Many of the inequalities here have analogues for the singular values of

non-Hermitian matrices (by exploiting the augmented matrix (2.80)). How-

ever, the situation is markedly different when dealing with eigenvalues of

non-Hermitian matrices; here, the spectrum can be far more unstable, if

pseudospectrum is present. Because of this, the theory of the eigenvalues of

a random non-Hermitian matrix requires an additional ingredient, namely

upper bounds on the prevalence of pseudospectrum, which after recenter-

ing the matrix is basically equivalent to establishing lower bounds on least

singular values. See Section 2.8.1 for further discussion of this point.

We will work primarily here with Hermitian matrices, which can be

viewed as self-adjoint transformations on complex vector spaces such as

Cn.

One can of course specialise the discussion to real symmetric matrices,

in which case one can restrict these complex vector spaces to their real

counterparts

Rn.

The specialisation of the complex theory below to the real

case is straightforward and is left to the interested reader.

1.3.1. Proof of spectral theorem. To prove the spectral theorem, it is

convenient to work more abstractly, in the context of self-adjoint operators

on finite-dimensional Hilbert spaces:

Theorem 1.3.1 (Spectral theorem). Let V be a finite-dimensional complex

Hilbert space of some dimension n, and let T : V → V be a self-adjoint

operator. Then there exists an orthonormal basis v1,...,vn ∈ V of V and

eigenvalues λ1,...,λn ∈ R such that Tvi = λivi for all 1 ≤ i ≤ n.

12Actually, this method eventually generates all of the eigenvalue inequalities, but this is a

non-trivial fact to prove; see [KnTaWo2004]