Page 147 - Textos de Matemática Vol. 44

ON ALMOST NORMAL MATRICES 137
and the so called ovular shape. The next theorem allows us to distinguish easily between the two.
Theorem 5.2. Let A be a pure almost normal 3-by-3 matrix, with a canonical form (1.1). Then W(A) is an ellipse if
λ 1 β 2 2 + λ 2 β 12 (5.3) μ = β12 + β2 ,
and has an ovular shape otherwise. Under condition (5.3) the ellipse W(A) has its foci at the eigenvalues λ1 , λ2 and the minor axis of the length β12 + β2 .
Proof. According to the ellipticity criterion from [7], a unitarily irreducible 3-by-3 matrix
λ1 x y 0 λ2 z 0 0 λ3
has an elliptical numerical range if and only if the number
(5.4)
λ1 |z|2 +λ2 |y|2 +λ3 |x|2 −xyz
|x|2 + |y|2 + |z|2
coincides with one of the eigenvalues λj. In our setting (and in our notation)
(5.4) simplifies to the right hand side of (5.3). Being a convex combination of λ1 and λ2 with positive coefficients, it cannot coincide with either of them (recall that λ1 ̸= λ2 due to (2.2)). Since μ is the only remaining eigenvalue, the ellipticity criterion boils down to (5.4). The description of W(A), provided that (5.4) holds, also follows from [7].
On the other hand, according to Theorem 5.1 the ovular shape is the only option left for W (A) if (5.4) fails.
Corollary 5.3. A pure almost normal 3-by-3 matrix cannot have a circular disk as its numerical range.
6. Aluthge transform
The Aluthge transform of a square matrix A is defined as
∆(A) = R1/2UR1/2,
where A = UR is the (right) polar representation of A. Recall that R =
(A∗A)1/2, and is therefore invertible for invertible A. Thus, for such A (6.1) ∆(A) = R1/2AR−1/2.