Antidiagonals of matrices in a unitary similarity orbit
In the context of this paper, an antidiagonal of a matrix is a vector of matrix elements, running from the first column to the first row, at right angles to the main diagonal. A given \(n \times n\) matrix has one antidiagonal of length \(k\) for each \(k = 1, 2,\dots,n\). For any antidiagonal \(v\), we can define \(s(v)\), the sum of its components, and \(p(v)\), the product of its components. Under the action of unitary similarity, the space of all \(n \times n\) complex matrices is partitioned into orbits. Correspondingly, antidiagonals assemble into sets, as do the values of the functions \(s\) and \(v\). The term orbit is not used because these sets can intersect nontrivially or even coincide for distinct orbits of matrices. Noting that, for a given orbit, these sets are identical and define the numerical range in the case \(k = 1\), the authors explore their geometry in the more complicated cases that \(2 \leq k \leq n\).
scientific article; zbMATH DE number 553966
scientific article