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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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