Second-order optimality condition for \({\Delta}H\)-matrices (Q1431665)

A \(\Delta H\) matrix \(A\) is a square matrix that can be written as \(A=D+DH-HD\), for certain Hermitian \(H\) and diagonal \(D\). \textit{A. Ruhe} [BIT 27, 585--598 (1987; Zbl 0636.15017)] gave a necessary and sufficient condition to solve the problem of finding a normal matrix \(N\) realizing the Frobenius distance from \(A\) to the variety of normal matrices. In the paper under review, the author gives a new characterization of the second-order optimality condition given by Ruhe, in terms of an eigenvalue problem.