On linear combinations of two tripotent, idempotent, and involutive matrices. (English) Zbl 1165.15022

Let (1) \(A = c_{1}A_{1} + c_{2}A_{2}\), where \(c_{1}, c_{2}\) are nonzero complex numbers, and (\(A_1, A_2\)) is a pair of two \(n\times n\) nonzero matrices. The purpose of this paper is mainly twofold: in case \(A_1\) and \(A_2\) are involutive matrices, to characterize all situations where a linear combination of the form (1) is a tripotent or an idempotent or an involutive matrix, then to determine all situations where a linear combination of the form (1) is an involutive matrix when \(A_1\) and \(A_2\) are tripotent or idempotent matrices.


15B57 Hermitian, skew-Hermitian, and related matrices
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[1] Baksalary, J.K.; Baksalary, O.M., Idempotency of linear combinations of two idempotent matrices, Linear algebra appl., 321, 3-7, (2000) · Zbl 0984.15021
[2] Özdemir, H.; Özban, A.Y., On idempotency of linear combinations of idempotent matrices, Appl. math. comput., 159, 439-448, (2004) · Zbl 1070.15009
[3] Baksalary, J.K.; Baksalary, O.M.; Styan, G.P.H., Idempotency of linear combinations of an idempotent matrix and a tripotent matrix, Linear algebra appl., 354, 21-34, (2002) · Zbl 1016.15027
[4] Benítez, J.; Thome, N., Idempotency of linear combinations of an idempotent and a t-potent matrix that commute, Linear algebra appl., 403, 414-418, (2005) · Zbl 1077.15022
[5] Baksalary, J.K.; Baksalary, O.M.; Özdemir, H., A note on linear combination of commuting tripotent matrices, Linear algebra appl., 388, 45-51, (2004) · Zbl 1057.15018
[6] H. Özdemir, M. Sarduvan, A.Y. Özban, N. Güler, On idempotency and tripotency of linear combinations of tripotent matrices, submitted for publication. · Zbl 1167.15019
[7] Baksalary, J.K.; Baksalary, O.M., When is a linear combination of two idempotent matrices the group involutory matrix?, Linear multilinear algebra, 54, 6, 429-435, (2006) · Zbl 1112.15009
[8] Rao, C.R.; Mitra, S.K., Generalized inverse of matrices and its applications, (1971), John Wiley New York
[9] Seber, G.A.F., Linear regression analysis, (1977), John Wiley New York · Zbl 0354.62055
[10] Graybill, F.A., Introduction to matrices with applications in statistics, (1969), Wadsworth Publishing Company Inc. California · Zbl 0188.51601
[11] Baldessari, B., The distribution of a quadratic form of normal random variables, Ann. math. statist., 38, 1700-1704, (1967) · Zbl 0155.27301
[12] Horn, R.A.; Johnson, C.R., Matrix analysis, (1985), Cambridge University Press Cambridge, UK · Zbl 0576.15001
[13] Meyer, C.D., Matrix analysis and applied linear algebra, (2000), SIAM Philadelphia
[14] Bethe, H.A.; Salpeter, E.E., Quantum mechanics of one- and two-electron atoms, (1977), Plenum Pub. Co. New York · Zbl 0089.21006
[15] Adler, S.L., Quaternionic quantum mechanics and quantum fields, (1995), Oxford University Press Inc. New York · Zbl 0885.00019
[16] Gromov, N.A., The matrix quantum unitary cayley – klein groups, J. phys. A, 26, L5-L8, (1993) · Zbl 0766.17012
[17] Ovchinnikov, M.A., Properties of viro – turaev representations of the mapping class group of a torus, J. math. sci. (NY), 113, 856-867, (2003) · Zbl 1040.57013
[18] Mestechkin, M.M., Restricted hartree – fock method instability, Int. J. quant. chem., 13, 469-481, (1978)
[19] Baksalary, O.M.; Benítez, J., Idempotency of linear combinations of three idempotent matrices, two of which are commuting, Linear algebra appl., 424, 320-337, (2007) · Zbl 1119.15025
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