Acta mathematica scientia, Series B >
SOME FURTHER NOTES ON THE MATRIX EQUATIONS ATXB + BTXTA = C ANDATXB + BTXA = C
Received date: 2013-11-06
Revised date: 2014-02-21
Online published: 2015-01-20
Supported by
This work was partially supported by FCT (Portugal) with national funds through Centro de Matem´atica da Universidade de Tr´as-os-Montes e Alto Douro (PEst-OE/MAT/UI4080/2014).
Dehghan and Hajarian, [4], investigated the matrix equations ATXB+BTXTA = C and ATXB + BTXA = C providing inequalities for the determinant of the solutions of these equations. In the same paper, the authors presented a lower bound for the product of the eigenvalues of the solutions to these matrix equations. Inspired by their work, we give some generalizations of Dehghan and Hajarian results. Using the theory of the numerical ranges, we present an inequality involving the trace of C when A,B,X are normal matrices satisfying ATB = BAT
Key words： matrix equation; eigenvalue; trace; permutation matrix
G.SOARES . SOME FURTHER NOTES ON THE MATRIX EQUATIONS ATXB + BTXTA = C ANDATXB + BTXA = C[J]. Acta mathematica scientia, Series B, 2015 , 35(1) : 275 -280 . DOI: 10.1016/S0252-9602(14)60156-9
[1] Goldberg M, Straus E G. Elementary inclusion relations of generalized numerical ranges. Linear AlgebraAppl, 1977, 18: 1–24
[2] Horn R A, Johnson C R. Matrix Analysis. Cambridge: Cambridge University Press, 1985
[3] Li C -K, Tsing Nam-Kiu. Linear operators that preserve the c-numerical range or radius of matrices. LinearMultilin Algebra, 1988, 23: 27–46
[4] Dehghan M, Hajarian M. A lower bound for the product of eigenvalues of solutions to matrix equations.Applied Mathematics Letters, 2009, 22: 1786–1788
[5] Kwon B H, Youn M J, Bien Z. On bounds of the Riccati and Lyapunov matrix equations. IEEE TransAutomat Control, 1985, 30: 1134–1135
[6] Komaroff N. Simultaneous eigenvalue lower bounds for the Lyapunov matrix equation. IEEE Trans AutomatControl, 1988, 33: 126–128
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