An extension of genralized pseudo-involutions

Solutions of the nonlinear matrix equation αA + βX = αβAX are characterized in terms of the Drazin inverse of A, and are shown to provide an extension of generalized pseudo-involutions when X = A_d. Computational techniques are developed to construct general n by n matrices A such that X = A_d, and corresponding complete sets of eigenvectors and associated generalized eigenvectors. These results are illustrated by numerical examples using real matrices and also integral matrices whose elements are chosen from a finite field.