Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników


2010 | 8 | 5 | 855-870

Tytuł artykułu

Equalities for orthogonal projectors and their operations


Treść / Zawartość

Warianty tytułu

Języki publikacji



A complex square matrix A is called an orthogonal projector if A 2 = A = A*, where A* denotes the conjugate transpose of A. In this paper, we give a comprehensive investigation to matrix expressions consisting of orthogonal projectors and their properties through ranks of matrices. We first collect some well-known rank formulas for orthogonal projectors and their operations, and then establish various new rank formulas for matrix expressions composed by orthogonal projectors. As applications, we derive necessary and sufficient conditions for various equalities for orthogonal projectors and their operations to hold.










Opis fizyczny




  • Central University of Finance and Economics


  • [1] Afriat S.N., Orthogonal and oblique projectors and the characteristics of pairs of vector spaces, Math. Proc. Cambridge Philos. Soc., 1957, 53(4), 800–816
  • [2] Anderson W.N.,Jr., Duffin R.J., Series and parallel addition of matrices, J. Math. Anal. Appl., 1969, 26(3), 576–594
  • [3] Baksalary J.K., Algebraic characterizations and statistical implications of the commutativity of orthogonal projectors, In: Proceedings of the 2nd International Tampere Conference in Statistics, University of Tampere, Tampere, Finland, 1987, 113–142
  • [4] Baksalary J.K., Baksalary O.M., Idempotency of linear combinations of two idempotent matrices, Linear Algebra Appl., 2000, 321(1–3), 3–7
  • [5] Baksalary J.K., Styan G.P.H., Around a formula for the rank of a matrix product with some statistical applications, In: Graphs, Matrices, and Designs: Festschrift in Honor of Norman J. Pullman, Lecture Notes in Pure and Applied Mathematics, 139, Marcel Dekker, New York, 1993, 1–18
  • [6] Ben-Israel A., Greville T.N.E., Generalized Inverses: Theory and Applications, 2nd ed., Springer, New York, 2003
  • [7] Benitez J., Rakocěvić V., Applications of CS decomposition in linear combinations of two orthogonal projectors, Appl. Math. Comput., 2008, 203(2), 761–769
  • [8] Benitez J., Thome N., Characterizations and linear combinations of k-generalized projectors, Linear Algebra Appl., 2005, 410, 150–159
  • [9] Benitez J., Thome N., k-group periodic matrices, SIAM J. Matrix Anal. Appl., 2006, 28(1), 9–25
  • [10] Bernstein D.S., Matrix Mathematics, 2nd ed., Princeton University Press, Princeton, 2009
  • [11] Berube J., Hartwig R.E., Styan G.P.H., On canonical correlations and the degrees of non-orthogonality in the threeway layout, In: Statistical Sciences and Data Analysis, Proceedings of the Third Pacific Area Statistical Conference, Makuhari (Chiba, Tokyo), Japan, December 11–13, 1991, VSP, Utrecht, 1993, 247–252
  • [12] Björck A., Numerical Methods for Least Squares Problems, SIAM, Philadelphia, 1996
  • [13] Campbell S.L., Meyer C.D.,Jr., Generalized Inverses of Linear Transformations, Dover, New York, 1991
  • [14] Cheng S., Tian Y., Moore-Penrose inverses of products and differences of orthogonal projectors, Acta Sci. Math. (Szeged), 2003, 69(3–4), 533–542
  • [15] De Moor B.L.R., Golub G.H., The restricted singular value decomposition: properties and applications, SIAM J. Matrix Anal. Appl., 1991, 12(3), 401–425
  • [16] Galperin A.M., Waksman Z., On pseudo-inverses of operator products, Linear Algebra Appl., 1980, 33, 123–131
  • [17] Greville T.N.E., Solutions of the matrix equation XAX = X, and relations between oblique and orthogonal projectors, SIAM J. Appl. Math., 1974, 26(4), 828–832
  • [18] Groß J., On the product of orthogonal projectors, Linear Algebra Appl., 1999, 289(1–3), 141–150
  • [19] Izumino S., The product of operators with closed range and an extension of the reverse order law, Tohoku Math. J., 1982, 34(1), 43–52
  • [20] Marsaglia G., Styan G.P.H., Equalities and inequalities for ranks of matrices, Linear and Multilinear Algebra, 1974/75, 2(3), 269–292
  • [21] Penrose R., A generalized inverse for matrices, Math. Proc. Cambridge Philos. Soc., 1955, 51(3), 406–413
  • [22] Rao C.R., Yanai H., General definition and decomposition of projector and some applications to statistical problems, J. Statist. Plann. Inference, 1979, 3(1), 1–17
  • [23] Spitkovsky I., Once more on algebras generated by two projections, Linear Algebra Appl., 1994, 208/209, 377–395
  • [24] Spitkovsky I.M., On polynomials in two projections, Electron. J. Linear Algebra, 2006, 15, 154–158
  • [25] Tian Y., The maximal and minimal ranks of some expressions of generalized inverses of matrices, Southeast Asian Bull. Math., 2002, 25(4), 745–755
  • [26] Tian Y., Rank equalities for block matrices and their Moore-Penrose inverses, Houston J. Math., 2004, 30(4), 483–510
  • [27] Tian Y., Using rank formulas to characterize equalities for Moore-Penrose inverses of matrix products, Appl. Math. Comput., 2004, 147(2), 581–600
  • [28] Tian Y., More on maximal and minimal ranks of Schur complements with applications, Appl. Math. Comput., 2004, 152(3), 675–692
  • [29] Tian Y., Problem 753: A rank identity, solved by M. Bataille and other seven solvers, College J. Math., 2004, 35, 230
  • [30] Tian Y., On mixed-type reverse-order laws for the Moore-Penrose inverse of a matrix product, Internat. J. Math. Math. Sci., 2004, 58, 3103–3116
  • [31] Tian Y., The equivalence between (AB)† = B †A † and other mixed-type reverse-order laws, Internat. J. Math. Ed. Sci. Tech., 2006, 37(3), 331–339
  • [32] Tian Y., Cheng S., The maximal and minimal ranks of A − BXC with applications, New York J. Math., 2003, 9, 345–362
  • [33] Tian Y., Styan G.P.H., Rank equalities for idempotent and involutory matrices, Linear Algebra Appl., 2001, 335, 101–117
  • [34] Tian Y., Styan G.P.H., A new rank formula for idempotent matrices with applications, Comment. Math. Univ. Carolin., 2002, 43(2), 379–384
  • [35] Tian Y., Takane Y., On sum decompositions of weighted least-squares estimators for the partitioned linear model, Comm. Statist. Theory Methods, 2008, 37(1–2), 55–69

Typ dokumentu



Identyfikator YADDA

JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.