Norm estimates for solutions of matrix equations AX-XB=C and X-AXB=C
Michael I. Gil'
Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 34 (2014), p. 191-206 / Harvested from The Polish Digital Mathematics Library
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Let A, B and C be matrices. We consider the matrix equations Y-AYB=C and AX-XB=C. Sharp norm estimates for solutions of these equations are derived. By these estimates a bound for the distance between invariant subspaces of matrices is obtained.

Publié le : 2014-01-01
EUDML-ID : urn:eudml:doc:270241 
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     author = {Michael I. Gil'},
     title = {Norm estimates for solutions of matrix equations AX-XB=C and X-AXB=C},
     journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
     volume = {34},
     year = {2014},
     pages = {191-206},
     zbl = {1312.15018},
     language = {en},
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Michael I. Gil'. Norm estimates for solutions of matrix equations AX-XB=C and X-AXB=C. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 34 (2014) pp. 191-206. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1163/

[000] [1] R. Bhatia, Matrix Analysis (Springer, NY, 1997). doi: 10.1007/978-1-4612-0653-8 | Zbl 0863.15001

[001] [2] R. Bhatia and P. Rosenthal, How and why to solve the operator equation AX-XB=Y, Bull. London Math. Soc. 29 (1997) 1-21. doi: 10.1112/S0024609396001828 | Zbl 0909.47011

[002] [3] R.R. Bitmead and H. Weiss, On the solution of the discrete time Lyapunov matrix equation in controllable canonical form, IEEE Trans. Automat. Control AC-24 (1979) 481-482. doi: 10.1109/TAC.1979.1102064 | Zbl 0404.93018

[003] [4] Yu. L. Daleckii and M.G. Krein, Stability of Solutions of Differential Equations in Banach Space, Amer. Math. Soc., Providence, R.I, 1971.

[004] [5] C. Davis and W. Kahan, The rotation of eigenvectors by a perturbation III, SIAM J. Numer. Anal. 7 (1970) 146-162. doi: 10.1137/0707001 | Zbl 0198.47201

[005] [6] M. Dehghan and M. Hajarian, The reflexive and anti-reflexive solutions of a linear matrix equation and systems of matrix equations, Rocky Mountain J. Math. 40 (3) (2010) 825-848. doi: 10.1216/RMJ-2010-40-3-825 | Zbl 1198.15011

[006] [7] B.W. Dickinson, Analysis of the Lyapunov equation using generalized positive real matrices, IEEE Trans. Autumut. Control AC-25 (1980) 560-563. doi: 10.1109/TAC.1980.1102391 | Zbl 0429.93031

[007] [8] M.I. Gil', Operator Functions and Localization of Spectra, Lecture Notes In Mathematics vol. 1830 (Springer-Verlag, Berlin, 2003). | Zbl 1032.47001

[008] [9] M.I. Gil', Difference Equations in Normed Spaces. Stability and Oscillations, North-Holland, Mathematics Studies 206 (Elsevier, Amsterdam, 2007).

[009] [10] M.I. Gil', Norm estimates for functions of two non-commuting matrices, Electronic Journal of Linear Algebra 22 (2011) 504-512. | Zbl 1223.15030

[010] [11] M.I. Gil', Matrix equations with diagonalizable coefficients, Gulf J. Math. 1 (2013) 98-104.

[011] [12] M.I. Gil, Bounds for spectra of operators on tensor of Euclidean spaces, PanAmerican Math. J. 24 (3) (2014) 35-47. | Zbl 1311.15025

[012] [13] I. Gohberg, R. Lancaster and L. Rodman, Invariant Subspaces of Matrices with Applications (Wiley, New York, 1986). | Zbl 0608.15004

[013] [14] L. Grubisic, N. Truhar and K. Veseli, The rotation of eigenspaces of perturbed matrix pairs, Linear Algebra and Appl. 436 (2012) 4161-4178. doi: 10.1016/j.laa.2012.01.026

[014] [15] J.Z. Hearon, Nonsingular solutions of TA-BT=C, Linear Algebra and Appl. 16 (1977) 5783. doi: 10.1016/0024-3795(77)90019-2

[015] [16] R. Horn and C. Johnson, Topics in Matrix Analysis (Cambridge University Press, Cambridge, 1991). doi: 10.1017/CBO9780511840371 | Zbl 0729.15001

[016] [17] M. Konstantinov, Da-Wei Gu, V. Mehrmann and P. Petkov, Perturbation Theory for Matrix Equations, Studies in Computational Mathematics, 9 (North Holland, Amsterdam, 2003). | Zbl 1025.15017

[017] [18] A.G. Mazko, Matrix Equations, Spectral Problems and Stability of Dynamic Systems. Stability, Oscillations and Optimization of Systems (Scientific Publishers, Cambridge, 2008). | Zbl 1152.93300

[018] [19] V. Ptak and N.J. Young, A generalization of the zero location theorem of Schur and Cohn, IEEE Trans. Automat. Control AC-25 (1980) 978-980. doi: 10.1109/TAC.1980.1102476 | Zbl 0463.93049