Idempotency of linear combinations of an idempotent matrix and a tripotent matrix. (English) Zbl 1016.15027

Let \(A\) be a matrix with entries in the complex numbers. Then \(A\) is called idempotent if \(A^2=A\), it is called tripotent if \(A^3=A.\) If \(B_1\) and \(B_2\) are nonzero idempotent matrices, then \(B=B_1-B_2\) is called essentially tripotent.
Suppose \(A\) is idempotent, \(B\) is essentially tripotent, and \(c_1,c_2\neq 0.\) The authors determine when a linear combination \(c_1A+c_2B\) is idempotent.


15B57 Hermitian, skew-Hermitian, and related matrices
62H10 Multivariate distribution of statistics
15A63 Quadratic and bilinear forms, inner products
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[1] Baksalary, J.K.; Baksalary, O.M., Idempotency of linear combinations of two idempotent matrices, Linear algebra appl., 321, 3-7, (2000) · Zbl 0984.15021
[2] Baldessari, B., The distribution of a quadratic form of normal random variables, Ann. math. statist., 38, 1700-1704, (1967) · Zbl 0155.27301
[3] Groß, J., On the product of orthogonal projectors, Linear algebra appl., 289, 141-150, (1999) · Zbl 0945.15015
[4] Halmos, P.R., Finite-dimensional vector spaces, (1958), Van Nostrand Princeton, NJ · Zbl 0107.01404
[5] Marsaglia, G.; Styan, G.P.H., Equalities and inequalities for ranks of matrices, Linear and multilinear algebra, 2, 269-292, (1974)
[6] Mathai, A.M.; Provost, S.B., Quadratic forms in random variables: theory and applications, (1992), Marcel Dekker New York · Zbl 0792.62045
[7] Rao, C.R.; Mitra, S.K., Generalized inverse of matrices and its applications, (1971), Wiley New York
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