Involutiveness of linear combinations of a quadratic‎ ‎or tripotent matrix and an arbitrary matrix

Document Type : Research Paper

Authors

1 College of Science‎, ‎Guangxi‎ ‎University for Nationalities‎, ‎Nanning 530006‎, ‎P. R‎. ‎China.

2 Departamento de Matematica Aplicada‎, ‎Instituto de Matematica Multidisciplinar‎, ‎Universidad Politecnica de Valencia‎, ‎Valencia 46022‎, ‎Spain.

3 College of Science‎, ‎Guangxi‎ ‎ University for Nationalities‎, ‎Nanning 530006‎, ‎P. R‎. ‎China.

Abstract

In this article, we characterize the involutiveness of the linear combination of the form
a1A1 +a2A2 when a1, a2 are nonzero complex numbers, A1 is a quadratic or tripotent matrix,
and A2 is arbitrary, under certain properties imposed on A1 and A2.

Keywords

Main Subjects


M. Aleksiejczyk and A. Smoktunowicz, On properties of quadratic matrices, Math. Pannon. 11 (2000), no. 2, 239--248.
J. K. Baksalary, O. M. Baksalary and G. P. H. Styan, Idempotency of linear combinations of an idempotent and a tripotent matrix, Linear Algebra Appl. 354 (2002) 21--34.
J. K. Baksalary, O. M. Baksalary and H. Özdemir, A note on linear combinations of commuting tripotent matrices, Linear Algebra Appl. 388 (2004) 45--51.
J. K. Baksalary and O. M. Baksalary, When is a linear combination of two idempotent matrices the group involutory matrix?, Linear Multilinear Algebra 54 (2006), no. 6, 429--435.
A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, Wiley-Interscience, 2nd edition, New York, 1974.
J. Bentez and N. Thome, fkg-group periodic matrices, SIAM J. Matrix Anal. Appl. 28 (2005), no. 1, 9--25.
J. Bentez, X. Liu and T. Zhu, Nonsingularity and group invertibility of linear combinations of two k-potent matrices, Linear Multilinear Algebra 58 (2010)), no. 7-8, 1023--1035.
D. S. Cvetkovic-Ilic and C. Y. Deng, The Drazin invertibility of the difference and the sum of two idempotent operators, J. Comput. Appl. Math. 233 (2010), no. 8, 1717--1722.
C. Y. Deng, On properties of generalized quadratic operators, Linear Algebra Appl. 432 (2010), no. 4, 847--856.
C. Y. Deng, Characterizations and representations of the group inverse involving idempotents, Linear Algebra Appl. 434 (2011), no. 4, 1067--1079.
C. Y. Deng, On the Drazin inverses involving power commutativity, J. Math. Anal. Appl. 378 (2011), no. 1, 314--323.
C. Y. Deng, D.S. Cvetkovic-Ilic, Y. Wei, On invertibility of combinations of k-potent operators, Linear Algebra Appl. 437 (2012), no. 1, 376--387.
C. Y. Deng, Characterizations of the commutators and the anticommutator involving idempotents, Appl. Math. Comput. 221 (2013) 351--359.
R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.
X. Liu and J. Bentez, The spectrum of matrices depending on two idempotents, Appl. Math. Lett. 24 (2011), no. 10, 1640--1646.
X. Liu, L. Wu and J. Bentez, On the group inverse of linear combinations of two group invertible matrices, Electron. J. Linear Algebra 22 (2011) 490--503.
X. Liu, L. Wu and Y. Yu, The group inverse of the combinations of two idempotent matrices, Linear Multilinear Algebra 59 (2011), no. 1, 101--115.
S. K. Mitra, P. Bhimasankaram and S. B Malik, Matrix Partial Orders, Shorted Operators And Applications, World Scientiic Publishing Co., Hackensack, 2010.
H. Özdemir and T. Petik, On the spectra of some matrices derived from two quadratic matrices, Bull. Iranian Math. Soc. 39 (2013), no. 2, 225--238.
H. Özdemir, M. Sarduvan, A. Y. Özban and N. Guler, On idempotency and tripotency of linear combinations of two commuting tripotent matrices, Appl. Math. Comput. 207 (2009), no. 1, 197--201.
C. R. Rao and S. K. Mitra, Generalized Inverse of Matrices and Its Applications, John Wiley & Sons, Inc., New York-London-Sydney, 1971.
M. Sarduvan and H. Özdemir, On linear combinations of two tripotent, idempotent, and involutive matrices, Appl. Math. Comput. 200 (2008), no. 1, 401--406.