40 1. Preparatory material
(1.52) and (1.54) are
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).
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
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
One can of course specialise the discussion to real symmetric matrices,
in which case one can restrict these complex vector spaces to their real
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]