Page 81 - Textos de Matemática Vol. 39

For each real t, define
λn⎠ .
0 Thus f(t) = σ2n−1Q(t). Given that
tI z2I − A
NEAREST TO A NORMAL MATRIX 71
2. The case A normal
Ikramov and Nazari in [2] extended the Malyshev’s formula (1.1) when A is a normal matrix, i.e. A∗A = AA∗, for the distance from A to the set of matrices X that have to z0 as an eigenvalue of multiplicity ≥ 3:
(2.1) ⎛z0I − A minm(z0,X)≥3 ∥X−A∥ = maxt1,t2≥0,z∈C σ3n−2 ⎝ 0
0
t1I z0I − A 0
zI ⎞ t2I ⎠ .
z0I − A
The proof was based in the possibility of reducing A to a diagonal matrix by means of a unitary matrix. In that proof I found ideas to prove the following theorem.
Theorem 2.1. With the notations in Section 1, let us suppose moreover that A is a normal matrix. If f(t) ≡ 0 or the maximum of f(t) is attained only at t = 0, then
(2.2) minz1,z2∈Λ(X)∥X − A∥ = max{σn(z1I − A), σn(z2I − A)}.
Proof. As A ∈ Cn×n is normal there exists a unitary matrix U ∈ Cn×n such
that
⎛⎜λ1 ⎞⎟ U∗AU = D = ⎜ λ2 ... ⎟
⎝ Q(t) := z1I − A
U 0 ∗ z1I − A tI U 0 = z1I − D tI 0U 0z2I−A0U 0z2I−D
f(t) = σ2n−1P(t). Besides, it is easy to see that
min X∈Cn×n ∥X−A∥=min Y∈Cn×n ∥Y −D∥. z1,z2∈Λ(X) z1,z2∈Λ(Y )
=: P (t),