Characterization of matrices using m-projectors and singular value decomposition in Minkowski space
محورهای موضوعی : Linear and multilinear algebra; matrix theory
1 - Department of School Education, HSS Razloo, Kund, Kulgam- 192221, JK, India
2 - Department of Higher Education, GDC(A.S.C), Srinagar-190008, JK, India
کلید واژه: singular value decomposition, Minkowski inverse, Range Symmetric, m-projectors, range disjoint, full range,
چکیده مقاله :
In this paper we characterize different classes of matrices in Minkowski space $\mathcal{M}$ by generalizing the singular value decomposition in terms of \emph{m}-projectors. Furthermore, we establish results on the relation between the range spaces and rank of the range disjoint matrices by utilizing the singular value decomposition obtained in terms of \emph{m}-projectors. Since there is no result on the formulation of Minkowski inverse of the sum of two matrices, we have established an expression for the Minkowski inverse of the sum of a range disjoint matrix with its Minkowski adjoint, which will ease to formulate an expression for the Minkowski inverse of the sum of two matrices in general case.
[1] K. Adem, Z. Zhour, The representation and approximation for the weighted minkowski inverse in Minkowski space, Math. Comput. Model. 47 (2007), 363-371.
[2] J. Baksalary, O. M. Baksalary, X. Liu, G. Trenkler, Further results of generalized and hypergeneralized projectors, Linear. Algebra. Appl. 429 (2008), 1038-1050.
[3] O. M. Baksalary, G. Trenkler, On disjoint range matrices, Linear. Algebra. Appl. 435 (2011), 1222-1240.
[4] A. Ben-isreal, T. Greville, Generalized Inverse: Theory and Applications, 2nd Edition, Springer Verlag, New York, 2003.
[5] I. Gohberg, P. Lancaster, L. Rodman, Indefinite Linear Algebra and Applications, Brikhauser Verlag, Basel, 2005.
[6] G. Golub, W. Kahan, Calculating the singular values and pseudo-inverse of a matrix, J. Soc. Industrial. Appl. Math. (Ser. B. Numerical Anal). 2 (1965), 205-224.
[7] G. H. Golub, C. Reinsch, Singular value decomposition and least square solutions, Numer. Math. 14 (1970), 403-420.
[8] R. E. Hartwig, Singular value decomposition and the moore-penrose inverse of bordered matrices, SIAM J. Appl. Math. 31 (1976), 31-41.
[9] R. E. Hartwig, K. Spindlebock, Matrices for which A ∗ and A † commute, Linear. Multilinear. Algebra. 14 (1984), 241-256.
[10] V. C. Klema, A. Laub, The singular value decomposition: its computation and some applications, Trans. Automatic. Cont. 25 (1980), 164-176.
[11] L. D. Lathauwer, B. D. Moor, J. Vandewalle, A multilinear singular value decomposition, SIAM J. Matrix Anal. Appl. 21 (2000), 1253-1278.
[12] A. A. Maciejewski, C. A. Klein, The singular value decomposition: computation and applications to robotics, Inter. J. Robotics. Res. 8 (1989), 63-79.
[13] G. Matsaglia, P. H. Styan, Equalities and inequalities for the rank of matrices, Linear. Multilinear. Algebra. 2 (1974), 269-292.
[14] M. Moonen, E. B. De Moor, SVD and Signal Processing, III. Algorithms, Applications and Architectures, Elsevier, Amsterdam, 1995.
[15] M. Renardy, Singular value decomposition in minkowski space, Linear. Algebra. Appl. 236 (1996), 53-58.
[16] M. Saleem Lone, D. Krishnaswamy, m-projetions involving minkowski inverse and range symmetric property in minkowski space, J. Linear. Topological. Algebra. 5 (2016), 215-228.
[17] M. Saleem Lone, D. Krishnaswamy, Representation of projectors involving minkowski inverse in Minkowski space, Indian J. Pure. Appl. Math. 48 (2017), 369-389.
[18] M. Schmidt, S. Rajagopal, Z. Ren, K. Moffat, Applications of singular valvue decomposition to the analysis of time resolved molecular x-ray data, Biophysical J. 84 (2002), 2112-2129.
[19] B. I. Shaini, F. Hoxha, Computing generalized inverses using matrix factorizations, Ser. Math. Inform. 28 (2013), 335-353.
[20] G. W. Stewart, the early history of the singular value decomposition, SIAM Review. 35 (1993), 551-566.
[21] R. E. Vaccaro, SVD and Signal Processing, II. Algorithms, Applications and Architectures, 1st Edition, Elsevier, Amsterdam, 1991.
[22] M. E. Wall, A. Rechtsteiner, L. M. Rocha, A practical approach to Microarray data analysis, 1st Edition, Springer, US, 2003.
[23] Z. Xing, On deterministic and non-deterministic muller matrix, J. Modern Opt. 39 (1992), 461-484.
[24] H. Yanai, K. Takeuchi, Y. Takane, Projection Matrices, Generalized Inverse Matrices and Singular Value Decomposition, New York, Springer Verlag, 2011.
[25] H. Zekraoui, Z. A. Zhour, C. Ozel, Some new algebraic and topological properties of Minkowski inverse in Minkowski space, Sci. World. J. 1 (2013), 1-6.
