Idempotency of linear combinations of two idempotent matrices. (English) Zbl 0984.15021

The authors give a complete solution for the problem of characterizing all situations, where a linear combination \(P=c_1P_1 +c_2P_2\) of two different idempotent matrices \(P_1\) and \(P_2\) is also an idempotent matrix. They point out a statistical interpretation of this idempotency problem.
Reviewer: T.Nono (Hiroshima)


15B57 Hermitian, skew-Hermitian, and related matrices
62H10 Multivariate distribution of statistics
Full Text: DOI


[1] J.K. Baksalary, Algebraic characterizations and statistical implications of the commutativity of orthogonal projectors, in: T. Pukkila, S. Puntanen (Eds.), Proceedings of the Second International Tampere Conference in Statistics, University of Tampere, Tampere, Finland, 1987, pp. 113-142
[2] Groß, J.; Trenkler, G., On the product of oblique projectors, Linear and multilinear algebra, 44, 247-259, (1998) · Zbl 0929.15016
[3] Mathai, A.M.; Provost, S.B., Quadratic forms in random variables: theory and applications, (1992), Dekker New York · Zbl 0792.62045
[4] Rao, C.R.; Mitra, S.K., Generalized inverse of matrices and its applications, (1971), Wiley New York
[5] Takane, Y.; Yanai, H., On oblique projectors, Linear algebra appl., 289, 297-310, (1999) · Zbl 0930.15003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.