On the product of oblique projectors. (English) Zbl 0929.15016

The paper shows that for two projectors (idempotents) \(P_1\) and \(P_2\), idempotency of the product \(P_1P_2\) is equivalent to the coincidence of the range of \(P_1P_2\) and a certain subspace, which only depends on the onto-and along-spaces of \(P_1\) and \(P_2\). Some further investigations are made, and necessary and sufficient conditions for the commutativity of two projectors are given.


15A27 Commutativity of matrices
Full Text: DOI


[1] Baksalary J. K., Algebraic characterizations and statistical implications of the commutativity of orthogonal projectors. Proceedings of the Second International Tampere Conference in Statistics (1987)
[2] Ben-Israel A., Generalized Inverses:Theory and Applications (1974)
[3] Brown, A. L. and Page, A. 1970. ”Elements of functional analysis”. NewYork: Van Nostrand. · Zbl 0199.17902
[4] DOI: 10.1080/03081087408817070
[5] Mitra S. K., A Raghu Raj Bahadur Festschrift pp 463– (1993)
[6] Rao C. R., Generalized Inverse of Matrices and its Applications (1971) · Zbl 0236.15004
[7] DOI: 10.1016/0024-3795(74)90023-8 · Zbl 0293.15006
[8] Takeuchi K., The Foundations of Multivariate Analysis (1982)
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.