Quantum Physics Division

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Idempotency of linear combinations of an idempotent matrix and a tripotent one

J. K. Baksalary^{1}
,
O. M. Baksalary^{2}
,
G. P. H. Styan^{3}

^{1} Department of Mathematics, Zielona Góra University, pl. Słowiański 9, PL-65-069, Zielona Góra, Poland

^{2} Department of Physics, Adam Mickiewicz University, ul. Umultowska 85, 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, 1-3, 21-34 (2001)**

**DOI:** 10.1016/S0024-3795(02)00343-9

Abstract:

The problem of characterizing situations, in which a linear combination C=c_{1}A+c_{2}B 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 B_{1} and B_{2} constituting its unique decomposition B=B_{1}−B_{2} are nonzero. The problem is considered also under the additional assumption that the differences A−B_{1} and A−B_{2} are Hermitian matrices. This obviously covers the case when A, B_{1}, and B_{2} are Hermitian themselves, when the problem can be interpreted from a statistical point of view.

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