On idempotency of linear combinations of idempotent matrices.(English)Zbl 1070.15009

Let $$P_1, P_2$$ and $$P_3$$ being any three different nonzero mutually commutative $$n\times n$$ idempotent matrices, and $$c_1,c_2$$ and $$c_3$$ being nonzero scalars, the problem of characterizing some situations, where a linear combination of the form $$P=c_1P_1+c_2P_2$$ or $$P=c_1P_1+c_2P_2+c_3P_3$$, is also an idempotent matrix is considered. Moreover two interesting results about the idempotency of linear combinations of $$2\times2$$ idempotent matrices and $$3\times3$$ idempotent matrices are given. A statistical interpretation of the idempotency problem considered in this study is also pointed out.

MSC:

 15A21 Canonical forms, reductions, classification 15A27 Commutativity of matrices 15A63 Quadratic and bilinear forms, inner products
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References:

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