Involutions and Generalized Centrosymmetric and Skew-centrosymmetric Matrices

We present new results and simple short proofs for known theorems about generalized centrosymmetric matrices and generalized skew-centrosymmetric (centroskew) matrices. We also make a comparison between the identity matrix and a nontrivial involutory matrix $K$ by studying the eigensystem of $K+M$, where $M$ is a structured matrix, with the goal of expressing this system entirely in terms of the corresponding eigensystem of $M$. Also, properties of some parametric families of the form $K+\rho M$ are studied.