Page 67 - Matrices theory and applications

3. Matrices with Real or Complex Entries
50
3.3.2 Applications
Theorem 3.3.3 Let H ∈ H n−1 , x ∈ CC
λ 1 ≤ ··· ≤ λ n−1 be the eigenvalues of H and µ 1 ≤ ··· ≤ µ n those of the
Hermitian matrix

H

H =
a
x ∗ x n−1 . ,and a ∈ IR be given. Let
One has then µ 1 ≤ λ 1 ≤ ··· ≤ µ j ≤ λ j ≤ µ j+1 ≤ ··· .
Proof
By Theorem 3.3.2, the inequality µ j ≤ λ j is obvious, because the infimum
is taken over a smaller set.
Conversely, let π : x → (x 1 ,... ,x n−1 ) T be the projection from CC n on
n
CC n−1 .If F is a linear subspace of CC of dimension j + 1, its image under
π contains a linear subspace G of dimension j (it will often be exactly of
dimension j). By Theorem 3.3.2, applied to H, one therefore has

R (F) ≥ R(G) ≥ λ j .
Taking the infimum, we obtain µ j+1 ≥ λ j .
The previous theorem is optimal, in the following sense.
Theorem 3.3.4 Let λ 1 ≤ ··· ≤ λ n−1 and µ 1 ≤ ··· ≤ µ n be real numbers
satisfying µ 1 ≤ λ 1 ≤ ··· ≤ µ j ≤ λ j ≤ µ j+1 ≤ ··· . Then there exist a vector
n
x ∈ IR and a ∈ IR such that the real symmetric matrix

Λ x
H = T ,
x a
where Λ = diag(λ 1 ,... ,λ n−1 ), has the eigenvalues µ j .
Proof
Let us compute the characteristic polynomial of H from Schur’s
5
complement formula (see Proposition 8.1.2):
T −1
p n (X)= X − a − x (XI n−1 − Λ) x det(XI n−1 − Λ)
 
x 2
j
=  X − a −  (X − λ j ).
X − λ j
j j
Let us assume for the moment that all the inequalities µ j ≤ λ j ≤ µ j+1
hold strictly. In particular, the λ j ’s are distinct. Let us consider the partial
fraction decomposition of the rational function

(X − µ l ) c j
l
= X − a − .
(X − λ j )
j j X − λ j
5
One may equally (exercise) compute it by induction on n.