Orthogonal tripotent matrices

Orthogonal tripotent matrices (i.e. A3=A=A∗) constitute a generalization of orthogonal idempotent matrices (i.e. A2=A=A∗). In this paper, we present different characterizations of orthogonal tripotent matrices in terms of matrix equations, integer powers of AA∗ and A∗A, averages involving A, A∗ and A†, as well as matrix rank and trace conditions. We further investigate the relationships between this class of matrices and other matrix classes, including normal matrices, EP matrices and partial isometry matrices, among others.