Page 68 - Matrices theory and applications

One thus obtains

a =
λ j ,
µ l −
l
j
a formula that could also have been found by comparing the traces of Λ
and of H. The inequalities λ j−1 <µ j <λ j ensure that each c j is positive,
because 3.4. The Spectrum and the Diagonal of Hermitian Matrices 51

(λ j − µ l )
l
(λ j − λ k )
c j = − .
k =j
√ √
Let us put, then, x j = c j (or −x j = c j ). We obtain, as announced,

p n (X)= (X − µ l ).
l
(m) (m)
In the general case one may choose sequences µ and λ that con-
l j
verge to the µ l ’s and the λ j ’s as m → +∞ and that satisfy the inequalities
in the hypothesis strictly. The first part of the proof (case with strict in-
equalities) provides matrices H (m) . Since the spectral radius is a norm over
Sym (IR) (the spectral radius is defined in the next Chapter), the sequence
n
(H (m) ) m∈IN is bounded. In other words, (a (m) ,x (m) ) remains bounded. Let
us extract a subsequence that converges to a pair (a, x) ∈ IR × IR n−1 .The
matrix H associated to (a, x) solves our problem, since the eigenvalues
depend continuously on the entries of the matrix.
Corollary 3.3.2 Let H ∈ Sym n−1 (IR) with eigenvalues λ 1 ≤ ··· ≤ λ n−1 .
Let µ 1 ,... ,µ n be real numbers satisfying µ 1 ≤ λ 1 ≤ ··· ≤ µ j ≤ λ j ≤
n
µ j+1 ≤ ··· . Then there exist a vector x ∈ IR and a ∈ IR such that the real
symmetric matrix

H x

H = T
x a
has the eigenvalues µ j .
The proof consists in diagonalizing H through an orthogonal conjugation,
then applying the theorem, and finally performing the inverse conjugation.

3.4 The Spectrum and the Diagonal of Hermitian
Matrices

Let us begin with an order relation between finite sequences of real num-
bers. If a =(a 1 ,... ,a n ) is a sequence of n real numbers, and if 1 ≤ l ≤ n,