On \(k\)-\(EP\) matrices
Let \(\mathbb{C}_{n\times n}\) be the space of \(n\times n\) complex matrices of order \(n\). A matrix \(A\in \mathbb{C}_{n\times n}\) is an EP matrix if \(N(A)= N(A^*)\), where \(N(A)\) is the null space and \(A^*\) is the conjugate transpose of \(A\) respectively [see \textit{I. J. Katz} and \textit{M. H. Pearl}, J. Res. Natl. Bur. Stand., Sect. B 70, 47-77 (1966; Zbl 0138.01401)]. Let \(k\) be a fixed product of disjoint transformations in \(S_n= \{1,2,\dots, n\}\). A matrix \(A\in \mathbb{C}_{n\times n}\) is \(k\)-Hermitian if \(a_{ij}= \overline{a}_{k(j), k(i)}\) for \(i,j\in \{1,2,\dots,n\}\) [see \textit{R. D. Hill} and \textit{S. R. Waters}, Linear Algebra Appl. 169, 17-29 (1992; Zbl 0757.15011)]. The authors introduce the concept of \(k\)-EP matrices as a generalization of \(k\)-Hermitian and EP matrices. Relations between \(k\)-EP and EP matrices are discussed and necessary and sufficient conditions are determined for a matrix to be \(k\)-EP.
scientific article; zbMATH DE number 1100771
scientific article