Page 85 - Textos de Matemática Vol. 39

NEAREST TO A NORMAL MATRIX 75
Therefore,
(2.9) lim σ2 Γ(t) = 0.
t→∞ Ifa1 =0,aswesawσ2Γ(t)≡0.
Case |a | = |a |. In this case, A = 0. The conclusions about the monotony 122
and asymptotic behavior of σ1 Γ(t) and σ2 Γ(t) , are the same, except for σ1Γ(0) = |a1| = σ2Γ(0).
Figure 1(b) shows an example.
Now we return to matrix R(t) in (2.3). We are interested in the behavior
of σ2n−1R(t) when t is close to 0. The singular values of Γk(0) are |z1 − λk| and |z2 − λk| without prejudging which of them is the greater one. Therefore, the singular values of R(0) are
|z1 −λ1|,|z2 −λ1|,...,|z1 −λk|,|z2 −λk|,...,|z1 −λn|,|z2 −λn|.
Hence, σ2n−1R(t) is the last but one of these values when ordering them from larger to smaller. Thinking about which of the eigenvalues of A is closest to z1 (respectively to z2), we distinguish two cases: Case 1 and Case 2.
Case 1. Let us suppose that there are i,j ∈ {1,...,n}, with i ̸= j, such that |z1 − λi| = min{|z1 − α|: α ∈ Λ(A)}
|z2 − λj | = min{|z2 − α| : α ∈ Λ(A)}.
The possibility that λi = λj is not excluded. It is well known that
(2.10) minz1,z2∈Λ(Y )∥Y − D∥ ≥ max{σn(z1I − D), σn(z2I − D)}.
Without loss of generality, let us suppose that |z1 − λi| ≤ |z2 − λj|. We will see that in this Case 1 the equality holds in (2.10). Define the matrix Y0 := D except the entries λi and λj that are replaced by z1 and z2, respectively. Then,
∥Y0−D∥=⎜ ⎜
...
⎟=|z2−λj|. ⎟
But,
σn(z1I−D)=|z1 −λi|,
σn(z2I−D)=|z2 −λj|;
⎛0 ⎜
...
⎞
⎟ ⎟ ⎟
⎜
0
⎜ ⎜
z1 − λi
0
⎟ ⎟ ⎟
⎜ ⎜ ⎜
0
z2 − λj
0
...
⎟ ⎟ ⎟
⎝
0
⎠