On Schur complement in \(\kappa\)-EP matrices
For a complex matrix \(M\) partitioned in the form \[ M=\begin{pmatrix} A & B \\ C & D \end{pmatrix} \] a Schur complement of \(A\) in \(M\) denoted by \((M/A)\) is defined as \(D-CA^{\text{-}}-D \) where \(A^{\text{-}}\) is a generalized inverse of \(A\). In this work necessary and sufficient conditions for \((M/A)\) to be \(\kappa\)-EP matrix are obtained for the cases \(\text{rank}(M)=\text{rank}(A)\) and \(\text{rank}(M)\neq \text{rank}(A)\) analogous to the results found in \textit{R. Penrose} [Proc. Cambridge Philos. Soc. 52, 17-19 (1956; Zbl 0070.12501)]. As an application, a decomposition of a partitioned matrix into complementary summands of \(\kappa\)-EP matrices is given (\(M_{1}\) and \(M_{2}\) are complementary summands of \(M\) if \(M=M_{1}+M_{2}\) and \(\text{rank}(M)=\text{rank}(M_{1})+\text{rank}(M_{2})\)).
scientific article; zbMATH DE number 1874474