m-Projections involving Minkowski inverse and range symmetric property in Minkowski space

Subject Areas : History and biography

M. Saleem Lone ¹ , D. Krishnaswamy ²

1 - Department of Mathematics, Annamalai University, Chidambaram, PO. Code 608002, Tamilnadu, India
2 - Department of Mathematics, Annamalai University, Chidambaram, PO. Code 608002, Tamilnadu, India

Received: 2016-07-19 Accepted : 2016-11-02 Published : 2016-11-07

Keywords: Minkowski inverse, m-projections, Range Symmetric, EP matrix,

Abstract :

In this paper we study the impact of Minkowski metric matrix on a projection in theMinkowski Space M along with their basic algebraic and geometric properties.The relationbetween the m-projections and the Minkowski inverse of a matrix A in the minkowski space Mis derived. In the remaining portion commutativity of Minkowski inverse in Minkowski SpaceM is analyzed in terms of m-projections as an analogous development and extension of theresults on EP matrices.

References:

[1] A. Ben-isreal, T. Greville, Generalized inverse: Theory and applications. New York, Springer Verlag, 2nd ed. 2003.

[2] A. R. Meenakshi, Range symmetric matrices in Minkowski space. Bulletin of Malaysian Math Sci Society, 23 (2000), 45-52.

[3] |, Generalized inverse of matrices in Minkowski space. Proc. Nat. Seminar Alg. Appln, 1 (2000), 1-14.

[4] A. R. Meenakshi and D. Krishnaswamy, Product of range symmetric block matrices in minkowski space. Bulletin of Malaysian Math Sci Society 29 ( 2006), 59-68.

[5] C. Schmoeger, Generalized projections in Banach algebra. Linear Algebra and its Applications, 430 (2009), 601-608.

[6] D. S. Djordevie, Product of EP operators on Hilbert space. Proc Amer Math Soc., 129 (2000), 1727-1731.

[7] D. S. Djordjevic, Y. Wei, Operators with equal projections related to their generalized inverse. Applied Mathematics and Computations, 155 (2004), 655-664.

[8] E. Boasso, On the moore penrose inverse, EP banach space operators and EP banach algebra elements. J Math. Anal. Appl., 339 (2008), 1003-1014.

[9] H. Schwerdtfeger, Introduction to linear algebra and the theory of matrices. Groningen, P. Noordhof, 1950.

[10] H. Seppo and N. Kenneth, On projections in a space with an indefinite metric. Linear Algebra and its Applications. 208/209 (1994), 401-417.

[11] H. K. Du and Y. Li, The spectral characterization of generalized projections. Linear Algebra Appl. 400 (2005), 313-318.

[12] I. J. Katz, Wigmann type theorems for EPr matrices. Duke Math J. 32 (1965), 423-428.

[13] I. Gohberg, P. Lancaster and L. Rodman, Indefinite linear algebra and applications. Basel, Boston, Berlin, Brikhauser verlag, 2005.

[14] J. Gro and K. Trenkler, Generalized and hypergeneralized projectors. Linear Algebra Appl. 364 (1997), 463-474.

[15] J. Gro, On the product of orthogonal projectors. Linear Algebra and its Applications, 289 (1999), 141-150.

[16] J. Rebaza, A first course in applied mathematics. New Jersey , Wiley 2012.

[17] J. J. Koliha, A simple proof of the product theorem for EP matrices. Linear Algebra and its Applications, 294 (1999), 213-215.

[18] J. K. Baksalary, O. M. Baksalary and X. Liu, Further properties of generalized and hypergeneralized projectors. Linear Algebra Appl. 389 (2004), 295-303.

[19] K. Adem, Z. A. Zhour, The representation and approximation for the weighted minkowski inverse in minkowski space. Mathematical and computer modeling, 47 (2008), 363-371.

[20] L. Lebtahi, N. Thome, A note on k-generalized projections. Linear Algebra Appl. 420 (2007), 572-575.

[21] M. Pearl, On normal and EPr matrices. Michigan Math J. 6 (1959), 1-5.

[22] M. Z. Petrovic, S. Stanimirovic, Representation and computation of f2; 3g and f2; 4g-inverses in indefinite inner product spaces. Applied Mathematics and Computation, 254 (2015), 157-171.

[23] O. Baksalary, G. Trenkler, Functions of orthogonal projectors involving the moore-penrose inverse. Comput. Math. Appl. 59 (2010), 764-778.

[24] O. M. Baksalary, G. Trenkler, Revisitation of the product of two orthogonal projectors. Linear Algebra and its Applications, 430 (2009), 2813-2833.

[25] P. R. Halmos, Finite-dimensional vector spaces. New York, Springer Verlag, 1974.

[26] R. Piziak, P. L. Odell and R. Hahn, Constructing projections on sums and intersections. Comput. Math. Appl. 37 (1999), 67-74.

[27] R. B. Bapat, J. S. Kirkland and K. M. Prasad, Combinatorial matrix theory and generalized inverses of matrices. New Delhi, Springer Verlag, 2013.

[28] R. E. Hartwig, I. J. Katz, On product of EP matrices. Linear Algebra and its Applications, 252 (1997), 339-345.

[29] R. Piziak et. al, Constructing projections on sums and intersections. Compt. Math. Appl. 37 ( 1999), 67-74.

[30] S. Cheng, Y. Tian, Two sets of new characterizations for normal and EP matrices. Linear Algebra and its Applications, 18 (1975), 327-333.

[31] S. L. Campbell, C. D. Meyer, EP operators and generalized inverse. Canad Math Bull., 18 (1975), 327-33.

[32] S. L. Campbell, C.D. Meyer, Generalized inverse of linear transformation. New York, Dover Publications, 1991.

[33] T. S. Baskett, I. J. Katz, Theorems on product of EPr matrices. Linear Algebra and its Applications, 2 (1969), 87-103.

[34] Y. Haruo, et al. Projection matrices, generalized inverse matrices and singular value decomposition. New York, Springer Verlag, 2011.

[35] Y. Tian, G. P. H. Styan, Rank equalities for idempotent and involutory matrices. Linear Algebra Appl. 335 (2001), 101-117.