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

Baksalary, Jerzy K.; Baksalary, Oskar Maria; Styan, George P. H.

Linear Algebra Appl. 354, No. 1-3, 21-34 (2002).

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.

Reviewer: Erich Ellers (Toronto / Ontario)

MSC:

15B57	Hermitian, skew-Hermitian, and related matrices
62H10	Multivariate distribution of statistics
15A63	Quadratic and bilinear forms, inner products

Keywords:

projector; quadratic form; \(\chi^2\) distribution; idempotency; idempotent matrix; tripotent matrix

References:

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