1.3. Eigenvalues and sums 39
1.3. Eigenvalues and sums of Hermitian matrices
Let A be a Hermitian n × n matrix. By the spectral theorem for Hermit-
ian matrices (which, for sake of completeness, we prove below), one can
diagonalise A using a
sequence11
λ1(A) . . . λn(A)
of n real eigenvalues, together with an orthonormal basis of eigenvectors
u1(A),...,un(A)
Cn.
The set {λ1(A),...,λn(A)} is known as the spec-
trum of A.
A basic question in linear algebra asks the extent to which the eigenval-
ues λ1(A),...,λn(A) and λ1(B),...,λn(B) of two Hermitian matrices A, B
constrain the eigenvalues λ1(A+B),...,λn(A+B) of the sum. For instance,
the linearity of trace
tr(A + B) = tr(A) + tr(B),
when expressed in terms of eigenvalues, gives the trace constraint
(1.52) λ1(A + B) + · · · + λn(A + B) = λ1(A) + · · · + λn(A)
+λ1(B) + · · · + λn(B);
the identity
(1.53) λ1(A) = sup
|v|=1
v∗Av
(together with the counterparts for B and A + B) gives the inequality
(1.54) λ1(A + B) λ1(A) + λ1(B),
and so forth.
The complete answer to this problem is a fascinating one, requiring a
strangely recursive description (once known as Horn’s conjecture, which is
now solved), and connected to a large number of other fields of mathemat-
ics, such as geometric invariant theory, intersection theory, and the combi-
natorics of a certain gadget known as a “honeycomb”. See [KnTa2001] for
a survey of this topic.
In typical applications to random matrices, one of the matrices (say, B) is
“small” in some sense, so that A+B is a perturbation of A. In this case, one
does not need the full strength of the above theory, and instead relies on a
simple aspect of it pointed out in [HeRo1995], [To1994], which generates
several of the eigenvalue inequalities relating A, B, and A + B, of which
11The eigenvalues are uniquely determined by A, but the eigenvectors have a little ambiguity
to them, particularly if there are repeated eigenvalues; for instance, one could multiply each
eigenvector by a complex phase eiθ. In this text we are arranging eigenvalues in descending order;
of course, one can also arrange eigenvalues in increasing order, which causes some slight notational
changes in the results below.
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