Page 83 - Textos de Matemática Vol. 39

and
Therefore, (2.4)
A1 +t2 √t4 +2A1t2 +A2 λ2F(t)= 2 − 2
.
NEAREST TO A NORMAL MATRIX
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dλ1F(t) = t + √ t3 + A1t
dt t4 +2A1t2 +A2
> 0 < 0
forallt>0,becauseA1 >0,A2 >0;
(2.5) dλ2F(t) = t − √ t3 + A1t
dt t4 +2A1t2 +A2
for all t > 0. The reason of this last inequality is that t<√ t3+A1t
is equivalent to
or But
t4 +2A1t2 +A2
t2(t4 + 2A1t2 + A2) < t6 + 2A1t4 + A21t2;
t6 +2A1t4 +A2t2 < t6 +2A1t4 +A21t2. A2 = |a1|4 + |a2|4 − 2|a1|2|a2|2,
A21 = |a1|4 + |a2|4 + 2|a1|2|a2|2.
Ifa1 =0,thenA2 =A21;henceλ2F(t)=0forallt≥0.Ifa1 ̸=0,then
0 < A2 < A21; therefore ∀t > 0, A2t2 < A21t2; which implies dλ2F(t) < 0, ∀t > 0.
√ dt Since the function x → x is increasing on 0 < x < ∞, we have that σ1 Γ(t)
is an increasing function and σ2Γ(t) is a decreasing function on [0,∞). This is illustrated by Figure 1(a); furthermore,
σ1Γ(0) = |a2| > |a1| = σ2Γ(0).
Now we will give two inequalities that are useful to see the asymptotic
behavior of the singular values of Γ(t) when t → ∞. First, (2.6) σ1Γ(t) ≥ |a2|2 + t2.
In fact,
σ1Γ(t) = maxx∈C2×1 ∥Γ(t)x∥2. ∥x∥2 =1