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Idempotency of linear combinations of three idempotent matrices, two of which are commuting

O. M. Baksalary^{1}
,
J. Benitez^{2}

^{1} Institute of Physics, Adam Mickiewicz University, ul. Umultowska 85, 61-614 Poznań, Poland

^{2} Departamento de Matemática Aplicada, Instituto de Matemática Multidisciplinar, Universidad Politécnica de Valencia, Camino de Vera s/n. 46022, Valencia, Spain

**Linear Algebra and Its Applications, 242, 320-337 (2007)**

**DOI:** 10.1016/j.laa.2007.02.016

Abstract:

The considerations of the present paper were inspired by Baksalary [O.M. Baksalary, Idempotency of linear combinations of three idempotent matrices, two of which are disjoint, Linear Algebra Appl. 388 (2004) 67–78] who characterized all situations in which a linear combination P=c_{1}P_{1}+c_{2}P_{2}+c_{3}P_{3}, with c_{i}, i=1,2,3, being nonzero complex scalars and P_{i}, i=1,2,3, being nonzero complex idempotent matrices such that two of them, P_{1} and P_{2} say, are disjoint, i.e., satisfy condition P_{1}P_{2}=0=P_{2}P_{1}, is an idempotent matrix. In the present paper, by utilizing different formalism than the one used by Baksalary, the results given in the above mentioned paper are generalized by weakening the assumption expressing the disjointness of P_{1} and P_{2} to the commutativity condition P_{1}P_{2}=P_{2}P_{1}.

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