8 1. SOME FUNDAMENTALS
S1. All the eigenvalues of A are real numbers, and thus can be ordered
as
λmin = λn ≤ λn−1 ≤ . . . ≤ λ1 = λmax.
Also there exist eigenvectors
x1,x2,...,xn
of A for λ1,λ2,...,λn,
respectively, that form an orthonormal basis of
n.
S2. For every nonzero vector x ∈
n,
λn ≤
xtAx
xtx
≤ λ1.
Equivalently, for every vector x ∈
n
of length 1, that is,
xtx
= 1,
λn ≤
xtAx
≤ λ1.
In fact,
λ1 =
max{xtAx
:
xtx
= 1} and λn =
min{xtAx
:
xtx
= 1}.
S3. More generally, each of the eigenvalues of A can be characterized
using subspaces of
n.
Let
x1,x2,...,xn
be an orthonormal basis
of
n
consisting of eigenvectors of A for λ1,λ2,...,λn, respectively.
Then for a nonzero vector y ∈
span{xi,...,xn}
λi ≥
ytAy
yty
with equality for y =
xi,
and for y ∈
span{x1,...,xi},
λi ≤
ytAy
yty
with equality for y =
xi.
Thus
λi = inf{sup
ytAy
yty
: y ∈ W : W a subspace of
n
of dimension n − i + 1}.
S4. (Interlacing of eigenvalues I.) Let Q be an n × m real matrix with
orthonormal columns
(QtQ
= Im), and let the eigenvalues of the
m × m matrix
QtAQ
be μ1 ≥ μ2 ≥ · · · ≥ μm. Then
λn−m+i ≤ μi ≤ λi (i = 1, 2,...,m).
S5. (Interlacing of eigenvalues II.) If μ1 ≥ μ2 ≥ · · · ≥ μm are the
eigenvalues of an m × m principal submatrix of A, then
λn−m+i ≤ μi ≤ λi (i = 1, 2,...,m).
In particular, if μ1 ≥ μ2 ≥ · · · ≥ μn−1 are the eigenvalues of an
n − 1 × n − 1 principal submatrix of A, then
λn ≤ μn−1 ≤ λn−1 ≤ · · · ≤ μ2 ≤ λ2 ≤ μ1 ≤ λ1.
