Page 80 - Textos de Matemática Vol. 39

70 JUAN-MIGUEL GRACIA
Theorem1.1. Letn≥2andletA∈Cn×n beanymatrixandz1,z2 ∈Cwith
z1 ̸= z2 . Then z1I − A tI min X∈Cn×n ∥X−A∥=maxt≥0σ2n−1 0 z2I−A ,
z1 ,z2 ∈Λ(X ) whenever the function
f (t) := σ2n−1 z1I − A tI , t ∈ R 0 z2I − A
is not identically zero and the maximum of f(t) is reached at some point t0 > 0. The function f satisfies f(−t) = f(t) for all real t, and
lim f(t) = 0. t→∞
I also made in [1] the following conjecture. Conjecture 1.2. If f(t) ≡ 0 or the maximum
maxt≥0 f(t) is attained only at t = 0, then
(1.2) min X∈Cn×n ∥X − A∥ = max{σn(z1I − A), σn(z2I − A)}. z1 ,z2 ∈Λ(X )
Lippert proved in [3] without restrictions on f, even in case of z1 = z2, that
(1.3) min X∈Cn×n ∥X−A∥=max{σn(z1I−A),σn(z2I−A),maxt≥0f(t)}. z1 ,z2 ∈Λ(X )
He used an ingenious argument of continuity, proving first (1.3) in the special case that: (1) z1 ̸= z2; (2) σn(z1I−A), σn−1(z1I−A), σn(z2I−A), σn−1(z2I−A) are all distinct and nonzero; (3) u∗nvn ̸= 0, where un, vn are the singular vectors to the smallest singular values of z1I −A or z2I −A. He then used the fact that the set formed by these triples (A, z1, z2) is dense in the space Cn×n × C × C.
In this article, I will prove Conjecture 1.2 when the matrix A is normal. I will need some additional notations. By σ(M) we denote the tuple
σ1(M),σ2(M),...,σmin(m,n)(M) and call it the singular spectrum of M. If N ∈ Cp×q and
σ(N) := σ1(N),σ2(N),...,σmin(p,q)(N), we define the union σ(M ) ∪ σ(N ) as the tuple
σ1(M),σ2(M),...,σmin(m,n)(M),σ1(N),σ2(N),...,σmin(p,q)(N),
where we repeat the singular values as many times as indicated by their mul- tiplicities. Thus, ∪ is not just the set union.