An alternative approach to characterize the commutativity of orthogonal projectors

Oskar Maria Baksalary; Götz Trenkler

Discussiones Mathematicae Probability and Statistics (2008)

  • Volume: 28, Issue: 1, page 113-137
  • ISSN: 1509-9423

Abstract

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In an invited paper, 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, [2], pp. 113-142] presented 45 necessary and sufficient conditions for the commutativity of a pair of orthogonal projectors. Basing on these results, he discussed therein also statistical aspects of the commutativity with reference to problems concerned with canonical correlations and with comparisons between estimators and between sets of linearly sufficient statistics corresponding to different linear models. In the present paper, parts of this analysis are resumed in order to shed some additional light on the problem of commutativity. The approach utilized is different than the one used by Baksalary, and is based on representations of projectors in terms of partitioned matrices. The usefulness of such representations is demonstrated by reinvestigating some of Baksalary's statistical considerations.

How to cite

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Oskar Maria Baksalary, and Götz Trenkler. "An alternative approach to characterize the commutativity of orthogonal projectors." Discussiones Mathematicae Probability and Statistics 28.1 (2008): 113-137. <http://eudml.org/doc/277013>.

@article{OskarMariaBaksalary2008,
abstract = {In an invited paper, 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, [2], pp. 113-142] presented 45 necessary and sufficient conditions for the commutativity of a pair of orthogonal projectors. Basing on these results, he discussed therein also statistical aspects of the commutativity with reference to problems concerned with canonical correlations and with comparisons between estimators and between sets of linearly sufficient statistics corresponding to different linear models. In the present paper, parts of this analysis are resumed in order to shed some additional light on the problem of commutativity. The approach utilized is different than the one used by Baksalary, and is based on representations of projectors in terms of partitioned matrices. The usefulness of such representations is demonstrated by reinvestigating some of Baksalary's statistical considerations.},
author = {Oskar Maria Baksalary, Götz Trenkler},
journal = {Discussiones Mathematicae Probability and Statistics},
keywords = {partitioned matrix; canonical correlations; ordinary least squares estimator; generalized least squares estimator; best linear unbiased estimator; commutativity; orthogonal projectors},
language = {eng},
number = {1},
pages = {113-137},
title = {An alternative approach to characterize the commutativity of orthogonal projectors},
url = {http://eudml.org/doc/277013},
volume = {28},
year = {2008},
}

TY - JOUR
AU - Oskar Maria Baksalary
AU - Götz Trenkler
TI - An alternative approach to characterize the commutativity of orthogonal projectors
JO - Discussiones Mathematicae Probability and Statistics
PY - 2008
VL - 28
IS - 1
SP - 113
EP - 137
AB - In an invited paper, 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, [2], pp. 113-142] presented 45 necessary and sufficient conditions for the commutativity of a pair of orthogonal projectors. Basing on these results, he discussed therein also statistical aspects of the commutativity with reference to problems concerned with canonical correlations and with comparisons between estimators and between sets of linearly sufficient statistics corresponding to different linear models. In the present paper, parts of this analysis are resumed in order to shed some additional light on the problem of commutativity. The approach utilized is different than the one used by Baksalary, and is based on representations of projectors in terms of partitioned matrices. The usefulness of such representations is demonstrated by reinvestigating some of Baksalary's statistical considerations.
LA - eng
KW - partitioned matrix; canonical correlations; ordinary least squares estimator; generalized least squares estimator; best linear unbiased estimator; commutativity; orthogonal projectors
UR - http://eudml.org/doc/277013
ER -

References

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  1. [1] W.N. Anderson Jr., E.J. Harner and G.E. Trapp, Eigenvalues of the difference and product of projections, Linear and Multilinear Algebra 17 (1985), 295-299. Zbl0593.15004
  2. [2] J.K. Baksalary, Algebraic characterizations and statistical implications of the commutativity of orthogonal projectors, pp. 113-142 in: Proceedings of the Second International Tampere Conference in Statistics, T. Pukkila, S. Puntanen (Eds.), University of Tampere, Tampere, Finland 1987. 
  3. [3] J.K. Baksalary, O.M. Baksalary and T. Szulc, A property of orthogonal projectors, Linear Algebra Appl. 354 (2002), 35-39. Zbl1025.15039
  4. [4] A. Ben-Israel and T.N.E. Greville, Generalized Inverses: Theory and Applications (2nd ed.), Springer-Verlag, New York 2003. Zbl1026.15004
  5. [5] R.E Hartwig and K. Spindelböck, Matrices for which A* and A † commute, Linear and Multilinear Algebra 14 (1984), 241-256. 
  6. [6] G. Marsaglia and G.P.H. Styan, Equalities and inequalities for ranks of matrices, Linear and Multilinear Algebra 2 (1974), 269-292. Zbl0297.15003
  7. [7] R. Piziak, P.L. Odell and R. Hahn, Constructing projections on sums and intersections, Comput. Math. Appl. 37 (1999), 67-74. Zbl0936.65053
  8. [8] Y. Tian and G.P.H. Styan, Rank equalities for idempotent and involutory matrices, Linear Algebra Appl. 335 (2001), 101-117. Zbl0988.15002

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