On generalized quadratic matrices

In this paper a wide class of matrices is considered, containing idempotent, involutory, nilpotent and several other types of matrices. Extending an approach considered by Radjavi and Rosenthal [H. Radjavi, P. Rosenthal, On commutators of idempotents, Linear Multilinear Algebra 50 (2) (2002) 121-124], we investigate the set Q(P) of square matrices A∈Cn×n satisfying the equation A2 = αA + βP for some complex numbers α and β and for some n × n nonzero complex idempotent matrix P such the AP = PA = A. Special attention is paid to the Moore-Penrose and group inverse of matrices belonging to Q(P).