Page 155 - Textos de Matemática Vol. 44

THE MAIN DIAGONAL PRODUCTS OF 3 × 3 NORMAL MATRICES WHOSE EIGENVALUES ARE THE THIRD ROOTS OF UNITY
HIROSHI NAKAZATO, ALEXANDER KOVACˇEC, NATA´LIA BEBIANO AND JOA˜O DA PROVIDEˆNCIA
Dedicated to the 60th birthday of Prof. Nata´lia Bebiano.
Abstract. The set of all possible products of the three diagonal entries of matrices in the unitary similarity orbit of a 3 × 3 diagonal matrix is discussed. The case that this matrix is defined by the third roots of unity is examined in detail and a curve lying in the set of products and apparently its boundary is given.
1. Introduction and the main theorem
In the paper [2], Bebiano et al. studied the k-th product numerical range
defined as
k
WkΠ(A) = { (UAU∗)ll : U ∈ U(n)}
l=1
for an n×n complex matrix A and 1 ≤ k ≤ n, where U(n) is the the group
of n × n unitary matrices. Especially they showed that if A is unitarily similar to 0m ⊕ B for m ≥ k, then WkΠ(A) is star-shaped with respect to 0. In the paper [10], the authors of this note investigated the unitary similarity orbit {UDU∗ : U ∈ U(3)} of a 3 × 3 complex diagonal matrix D = diag(λ1, λ2, λ3) and the related k-th product numerical range WkΠ(D) for k = 2, 3, defined as
(1.1) W2Π(D) = {(UDU∗)11(UDU∗)22 : U ∈ U(3)}, (1.2) W3Π(D) = {(UDU∗)11(UDU∗)22(UDU∗)33 : U ∈ U(3)}.
The matrix UDU∗ is a normal matrix. The description of such sets is closely re- lated with the set of unistochastic matrices which also appear in mathematical physics ([3], [4], [5], [6]). In the paper [10], p. 712–714 the authors determined
2010 Mathematics Subject Classification. Primary 15B57; Secondary 15A60
Key words and phrases. Almost normal matrix, numerical range, Aluthge transform
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