A generalization of weighted versions of the determinant, permanent and the generalized inverse of rectangular matrices
الموضوعات :
1 - Department of Mathematics, Zanjan Branch, Islamic Azad University, P.O. Box 100190, Zanjan, Iran
الکلمات المفتاحية: The generalized weighted determinant, the generalized weighted permanent, the generalized Cauchy-Binet formula, the generalized Laplace expansion formula, the generalized determinantal inverse, Moore-Penrose weighted inverse,
ملخص المقالة :
In this paper, we first generalized the weighted versions of determinants, permanents and the generalized inverses of rectangular matrices. We also investigate some of their algebraic properties. As a by product of the above investigation, we then present a determinantal representation for the general and Moore-Penrose inverses which satisfy on certain conditions. Finally, we give a general algorithm for determining the inverse of some certain class of the rectangular matrices defined based on weighted determinants.
[1] P. S. Abhimanyu, Defining the determinant-like function for m by n matrices using the exterior algebra, Adv. Appl. Clifford Algebras. 23 (2013), 787-792.
[2] E. Arghiriade, A. Dragomir, Une nouvelle definition de l’inverse generalisée d’une matrice, Rendiconti Lincei. Scienze Fisiche e Naturali. Serie XXXV. 35 (1963), 158-165.
[3] M. Bayat, A bijective proof of generalized Cauchy-Binet, Laplace, Sylvester and Dodgson formulas, Linear. Multilinear. Algebra., In press.
[4] A. Ben-Israel, Generalized inverses of matrices: a perspective of the work of Penrose, Math. Proc. Camb. Phil. Soc. 100 (1986), 407-425.
[5] C. E. Cullis, Matrices and Determinoids, Cambridge University Press, Vol 3, 1925.
[6] R. Gabril, Extinderea Complementilor Algebrici Generalizati la Matrici Oarecare, Studii si Cercetari Matematice. 17-Nr. 10 (1965), 1566-1581.
[7] V. N. Joshi, A Determinant for Rectangular Matrices, Bull. Austral. Math. Soc. 21 (1980), 137-146.
[8] H. Minc, Permanents, Encyclopedia of Mathematics and its Applications, 6, Addison-Wesley, Reading, Mass, 1978.
[9] Y. Nakagami, H. Yanai, On Cullis determinant for rectangular matrices, Linear. Algebra. Appl. 422 (2007), 422-441.
[10] R. Penrose, A generalized inverse for matrices, Proc. Cambridge Philos Soc. 51 (1955), 406-413.
[11] M. Radic, A definition of determinant of rectangular matrix, Glasnik Matematicki. Ser III. 1 (21) (1966), 17-22.
[12] M. Radic, Areas of certain polygons in connection with determinants of rectangular matrices, Beiträge Algebra Geom. 49 (1) (2008), 71-96.
[13] M. Radic, Inverzije pravokutnih matrica, Doktorska disertacije, 1964.
[14] M. Radic, Some contribution to the inversion of rectangular matricrs, Glasnik Matematicki. Ser III. 1 (21) (1966), 23-37.
[15] M. Radic, R. Susanj, Geometrical meaning of one generalization of the determinant of a square matrix, Glasnik Matematicki. Ser III. 29 (49) (1994), 217-233.
[16] P. Stanimirovic, M. Stankovic, Determinants of rectangular matrices and the Moore-Penrose inverse, Novi Sad J. Math, Vol. 27 (1) (1997), 53-69.
[17] M. Stojakovic, Determinante Nekvadratnih Matrica, Vesnik DMNRS. 1-2 (1952), 9-12.
[18] L. Tan, Signs in the Laplace expansions and the parity of the distinguished representatives, Discrete Math. 131 (1994), 287-299.
