Some rank equalities for finitely many tripotent matrices

Document Type : Research Paper

Authors

Sakarya University, Department of Mathematics, Sakarya, 54187, Turkey.

Abstract

‎A rank equality is established for the sum of finitely many tripotent matrices via elementary block matrix operations‎. ‎Moreover‎, ‎by using this equality and Theorems 8 and 10 in [Chen M‎. ‎and et al‎. ‎On the open problem related to rank equalities for the sum of finitely many idempotent matrices and its applications‎, ‎The Scientific World Journal 2014 (2014)‎, ‎Article ID 702413‎, ‎7 pages‎.]‎, ‎some other rank equalities for tripotent matrices are given‎. ‎Furthermore‎, ‎we obtain several rank equalities related to some special types of matrices‎, ‎some of which are available in the literature‎, ‎from the results established‎.

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J. Bentez, M. Sarduvan, S. Ulker and H. Ozdemir, On nonsingularity of combinations of three group invertible matrices and three tripotent matrices, Linear Multilinear Algebra 61 (2013), no. 4, 463--481.
M. Chen, Q. Chen, Q. Li and Z. Yang, On the open problem related to rank equalities for the sum of finitely many idempotent matrices and its applications, The Scientific World Journal 2014 (2014), Article ID 702413, 7 pages.
F.A. Graybill, Matrices with Applications in Statistics, Wadsworth International Group, 2nd edition, Belmont, California, 1983.
D.A. Harville, Matrix Algebra From a Statistician's Perspective, Springer-Verlag, New York, 1997.
G. Marsaglia and G.P.H. Styan, Equalities and inequalities for ranks of matrices, Linear Multilinear Algebra 2 (1974/75), no. 3, 269--292.
T. Petik, M. Uc and H. Ozdemir, Generalized quadraticity of linear combination of two generalized quadratic matrices, Linear Multilinear Algebra 63 (2015), no. 12, 2430--2439.
C.R. Rao and S.K. Mitra, Generalized Inverse of Matrices and Its Applications, John Wiley & Sons Inc. New York-London-Sydney, 1971.
Y. Tian and G.P.H. Styan, Rank equalities for idempotent and involutory matrices, Linear Algebra Appl. 335 (2001) 101--117.
Y. Tian and G.P.H. Styan, A new rank formula for idempotent matrices with applications, Comment. Math. Univ. Carolin. 43 (2002), no. 2, 379--384.
Y. Tian and G.P.H. Styan, Rank equalities for idempotent matrices with applications, J. Comput. Appl. Math. 191 (2006), no. 1, 77--97.
M. Uc, T. Petik, and H. Ozdemir, The generalized quadraticity of linear combinations of two commuting quadratic matrices, Linear Multilinear Algebra 64 (2016), no. 9, 1696--1715.
K. Zuo, Some rank equalities about combinations of two idempotent matrices, Wuhan Univ. J. Nat. Sci. 15 (2010), no. 5, 380--384.
K. Zuo, Nonsingularity of the difference and the sum of two idempotent matrices, Linear Algebra Appl. 433 (2010), no. 2, 476--482.