Idempotency of linear combinations of an idempotent matrix and a tripotent matrix
J. K. Baksalary1 , O. M. Baksalary2 , G. P. H. Styan3
1 Faculty of Mathematics, Informatics, and Econometrics, Zielona Góra University, ul. Podgórna 50, PL 65-246 Zielona Góra, Poland
2 Department of Physics, A. Mickiewicz University, 61-614 Poznań, Poland
3 Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, QC, Canada H3A 2K6
Linear Algebra and Its Applications, 354, 21-34 (2002)
The problem of characterizing situations, in which a linear combination C=c1A+c2B of an idempotent matrix A and a tripotent matrix B is an idempotent matrix, is thoroughly studied. In two particular cases of this problem, when either B or −B is an idempotent matrix, a complete solution follows from the main result in [Linear Algebra Appl. 321 (2000) 3]. In the present paper, a complete solution is established in all the remaining cases, when B is an essentially tripotent matrix in the sense that both idempotent matrices B1 and B2 constituting its unique decomposition B=B1−B2 are nonzero. The problem is considered also under the additional assumption that the differences A−B1 and A−B2 are Hermitian matrices. This obviously covers the case when A, B1, and B2 are Hermitian themselves, when the problem can be interpreted from a statistical point of view.