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**Unformatted text preview: **1 ≥ (x∗ Ax/x∗ x) ≥ λn for all x = 0. The term x∗ Ax/x∗ x
is referred to as a Rayleigh quotient in honor of the famous English physicist
John William Strutt (1842–1919) who became Baron Rayleigh in 1873.
It’s only natural to wonder if the intermediate eigenvalues of a hermitian
matrix have representations similar to those for the extreme eigenvalues as described in (7.5.4). Ernst Fischer (1875–1954) gave the answer for matrices in 1905,
and Richard Courant (1888–1972) provided extensions for inﬁnite-dimensional
operators in 1920. D
E Courant–Fischer Theorem The eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λn of a hermitian matrix An×n are
λi = max dim V =i min x∗ Ax and x∈V
x 2 =1 λi = min dim V =n−i+1 T
H max x∗ Ax.
x∈V
x 2 =1 (7.5.5) When i = 1 in the min-max formula and when i = n in the maxmin formula, V = C n , so these cases reduce to the equations in (7.5.4).
Alternate max-min and min-max formulas are given in Exercise 7.5.12. IG
R Proof. Only the min-max characterization is proven—the max-min proof is
analogous (Exercise 7.5.11). As shown in Example 7.5.1, a change of coordinates
y = U∗ x with a unitary U such that U∗ AU = D = diag (λ1 , λ2 , . . . , λn ) has
the eﬀect of replacing A by D, so we need only establish that Y
P
λi = min dim V =n−i+1 O
C max y∗ Dy.
y∈V
y 2 =1 For a subspace V of dimension n − i + 1, let SV = {y ∈ V , y
SV = {y ∈ V ∩ F , y 2 = 1}, where 2 = 1}, and let F = span {e1 , e2 , . . . , ei } . Note that V ∩ F = 0, for otherwise dim(V + F ) = dim V + dim F = n + 1,
which is impossible. In other words, SV contains those vectors of SV of the
i
2
form y = (y1 , . . . , yi , 0, . . . , 0)T with
j =1 |yj | = 1. So for each subspace V
with dim V = n − i + 1,
i y∗ Dy = i λj |yj |2 ≥ λi
j =1 |yj |2 = λi for all y ∈ SV . j =1 Since SV ⊆ SV , it follows that maxSV y∗ Dy ≥ maxSV y∗ Dy ≥ λi , and hence
min max y∗ Dy ≥ λi .
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