Page 82 - Textos de Matemática Vol. 39

72 JUAN-MIGUEL GRACIA
Now we look at the matrix P(t) in detail,
⎛⎜ z 1 − λ 1 t ⎞⎟ ⎜ z1−λ2 t ⎟ ⎜... ...⎟
P (t) = ⎜ z1 − λn t ⎟
⎜ z2 −λ1 ⎜
⎜⎝
⎟ ⎟ ⎟⎠
z2 − λ2
By means of similarity permutations the matrix P(t) is unitarily similar to
. . .
(2.3) where
R(t) := Γ1(t) ⊕ · · · ⊕ Γn(t), z1 − λk t
Γk(t):=
0 z2 −λk , k=1,...,n.
So, f(t) = σ2n−1R(t). Besides, for the singular spectra we have σQ(t) = σR(t) = σΓ1(t) ∪ · · · ∪ σΓn(t).
Each block is of the form
Γk(t) := z1 − λk t 0 z2 − λk
Γ(t) := a1 t , with a1,a2 ∈ C; 0 a2
furthermore a1 ̸= a2 because z1 ̸= z2. Next, we are going to prove that the functions σ1Γ(t) and σ2Γ(t) are monotone when t varies in [0,∞).
Case |a1| < |a2|. For all real t,
F(t):=Γ∗(t)Γ(t)=|a1|2 0 +t0 a¯1+t20 0. 0|a2|2 a10 01
So,
As F (t) is a Hermitian matrix, its eigenvalues are real numbers; hence, we can
F(t)=|a1|2 a¯1t . a1t |a2|2+t2
suppose they are ordered λ1F(t) ≥ λ2F(t). For t = 0,
F(0) = |a1|2 0 , which implies λ F(0) = |a |2,λ F(0) = |a |2.
2 2 1
For each t ∈ R, putting A1 := |a1|2 + |a2|2 and A2 := |a1|2 − |a2|22, we
0 |a2|2 1
obtain A1 +t2 √t4 +2A1t2 +A2 λ1F(t)= 2 + 2
z2 − λn