Norm estimates for solutions of matrix equations AX-XB=C and X-AXB=C

Michael I. Gil'

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2014)

  • Volume: 34, Issue: 2, page 191-206
  • ISSN: 1509-9407

Abstract

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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.

How to cite

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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 34.2 (2014): 191-206. <http://eudml.org/doc/270241>.

@article{MichaelI2014,
abstract = {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.},
author = {Michael I. Gil'},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {matrix equations; norm estimates; perturbations; invariant subspaces},
language = {eng},
number = {2},
pages = {191-206},
title = {Norm estimates for solutions of matrix equations AX-XB=C and X-AXB=C},
url = {http://eudml.org/doc/270241},
volume = {34},
year = {2014},
}

TY - JOUR
AU - Michael I. Gil'
TI - Norm estimates for solutions of matrix equations AX-XB=C and X-AXB=C
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2014
VL - 34
IS - 2
SP - 191
EP - 206
AB - 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.
LA - eng
KW - matrix equations; norm estimates; perturbations; invariant subspaces
UR - http://eudml.org/doc/270241
ER -

References

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  1. [1] R. Bhatia, Matrix Analysis (Springer, NY, 1997). doi: 10.1007/978-1-4612-0653-8 Zbl0863.15001
  2. [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 Zbl0909.47011
  3. [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 Zbl0404.93018
  4. [4] Yu. L. Daleckii and M.G. Krein, Stability of Solutions of Differential Equations in Banach Space, Amer. Math. Soc., Providence, R.I, 1971. 
  5. [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 Zbl0198.47201
  6. [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 Zbl1198.15011
  7. [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 Zbl0429.93031
  8. [8] M.I. Gil', Operator Functions and Localization of Spectra, Lecture Notes In Mathematics vol. 1830 (Springer-Verlag, Berlin, 2003). Zbl1032.47001
  9. [9] M.I. Gil', Difference Equations in Normed Spaces. Stability and Oscillations, North-Holland, Mathematics Studies 206 (Elsevier, Amsterdam, 2007). 
  10. [10] M.I. Gil', Norm estimates for functions of two non-commuting matrices, Electronic Journal of Linear Algebra 22 (2011) 504-512. Zbl1223.15030
  11. [11] M.I. Gil', Matrix equations with diagonalizable coefficients, Gulf J. Math. 1 (2013) 98-104. 
  12. [12] M.I. Gil, Bounds for spectra of operators on tensor of Euclidean spaces, PanAmerican Math. J. 24 (3) (2014) 35-47. Zbl1311.15025
  13. [13] I. Gohberg, R. Lancaster and L. Rodman, Invariant Subspaces of Matrices with Applications (Wiley, New York, 1986). Zbl0608.15004
  14. [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 
  15. [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 
  16. [16] R. Horn and C. Johnson, Topics in Matrix Analysis (Cambridge University Press, Cambridge, 1991). doi: 10.1017/CBO9780511840371 Zbl0729.15001
  17. [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). Zbl1025.15017
  18. [18] A.G. Mazko, Matrix Equations, Spectral Problems and Stability of Dynamic Systems. Stability, Oscillations and Optimization of Systems (Scientific Publishers, Cambridge, 2008). Zbl1152.93300
  19. [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 Zbl0463.93049

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