Page 145 - Textos de Matemática Vol. 44

ON ALMOST NORMAL MATRICES 135
Theorem 4.1. Let A be an almost normal matrix. Then rank[A] = min{3, k},
where k is the PAN-rank of A.
Proof. Since [A] = 0⊕[Aa], it suffices to consider pure almost normal matrices only. They are not normal, so the rank of their self-commutators is between 2 and n, which proves the statement for n = 2. It remains to show that for a pure almost normal A with n ≥ 3 the rank of the self-commutator equals 3.
A direct computation reveals that the self-commutator of (1.1) equals
 
(βiβj)n−1 i,j=1
(μ−λ1)β1  . 
.
With β1 ̸= 0, multiplication of the latter matrix on the left by the triangular invertible matrix
1 [0 β2 ... βn−1 μ−λ1]T[1 0...0]−I β1
yields
β12 β1β2 ... β1βn−1 (μ−λ1)β1 
00 ... 0 (λ1 −λ2)β2  .. . .
− 
(μ−λn−1)βn−1 . (μ−λ1)β1 ... (μ−λn−1)βn−1 −(β12 +···+βn2−1)
.. . ..

0 0
... 0 (λ−λ)β 1 n−1 n−1
. . . (λ1 − λn−1)βn−1
row form a basis of the row space of [A]. Thus, rank[A] = 3.
5. Numerical range
The numerical range (also known as the field of values) of A ∈ Cn×n is
defined as
W(A)={⟨Ax,x⟩:x∈Cn, ∥x∥=1}
(here of course ⟨.,.⟩ stand for the usual scalar product on Cn, and ∥.∥ for the norm associated with it).
This is an extensively studied object, see e.g. [5]. In particular, it is known that W(A) is a compact convex subset of C containing the spectrum of A, and thus the convex hull of the spectrum as well:
(5.1) W (A) ⊇ conv σ(A).
0 (λ1−λ2)β2
Under conditions (2.2) and with n ≥ 3, the first, last, and any one additional
∗