On the Open Problem Related to Rank Equalities for the Sum of Finitely Many Idempotent Matrices and Its Applications

Abstract

Tian and Styan have shown many rank equalities for the sum of two and three idempotent matrices and pointed out that rank equalities for the sumP1+⋯+PkwithP1,…,Pkbe idempotent (k>3) are still open. In this paper, by using block Gaussian elimination, we obtained rank equalities for the sum of finitely many idempotent matrices and then solved the open problem mentioned above. Extensions to scalar-potent matrices and some related matrices are also included.