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136 T. MORAN AND I. SPITKOVSKY
Moreover, the numerical range is invariant under unitary similarities. If A is unitarily similar say to A1 ⊕ A2, then W(A) = conv{W(A1),W(A2)}. This implies in particular that, for normal matrices, the equality holds in (5.1), so in this case W(A) is a polygon. For almost normal A, in turn,
W (A) = conv{W (An), W (Aa)} = conv{σ(An), W (Aa)},
according to the decomposition (2.1). Consequently, only the case of pure al-
most normal matrices is of interest.
5.1. Roundness of W(A). Being unitarily irreducible, pure almost normal matrices cannot have sharp points on the boundary of their numerical range, since every such point is a normal eigenvalue. Flat portions on the boundary, however, are potentially possible for unitarily irreducible matrices, starting with n = 3, see e.g. [7]. Therefore it is not trivial to observe that they do not materialize in the case of pure almost normal matrices of any size.
Theorem 5.1. Let A be a pure almost normal matrix. Then the boundary of W(A) is a smooth curve with positive finite curvature.
Proof. According to [4], it suffices to show that for any θ ∈ R the maximal eigenvalue of Re(eiθ A) is simple. Since the matrices eiθ A are pure almost normal along with A, we need only to prove the claim for θ = 0. But, for A given by (1.1),
ξ1 0 ... 0 β1/2
(5.2) ReA= .
0 0
. β n − 1 / 2
0 ξ2 .
...
. . . . . .
0
ξ n − 1 βn−1/2
β2/2 .
β1/2 β2/2
ν
(hereξj =Reλj,j=1,...,n−1;ν=Reμ).
Due to the interlacing eigenvalues theorem, multiple eigenvalues of (5.2),
if any exist, must coincide with some ξj. On the other hand, ξj is not an eigenvalue (and thus not an endpoint of the numerical range) of the 2-by-2 block of (5.2) located in its j, n row and column, since βj ̸= 0. Consequently, for all j = 1,...,n−1, ξj are not endpoints of W(ReA), and therefore not the extremal eigenvalues of Re A.
5.2. The 3-by-3 case. According to Kippenhahn’s classification ([8], see also [9]), for n = 3 there are three possible shapes of W (A) for unitarily irreducible A, and one of these types has a flat portion on the boundary. This leaves two other options available for pure almost normal matrices: the elliptical shape
