Analytic perturbation of Sylvester and Lyapunov matrix equations

We consider an analytic perturbation of the Sylvester matrix equation. Mainly we are interested in the singular case, that is, when the null space of the unperturbed Sylvester operator is not trivial, but the perturbed equation has a unique solution. In this case, the solution of the perturbed equation can be given in terms of a Laurent series. We provide a necessary and sufficient condition for the existence of a Laurent series with a first order pole. An efficient recursive procedure for the calculation of the Laurent series' coefficients is given. Finally, we show that in the particular, but practically important case of semisimple eigenvalues, the recursive procedure can be written in a compact matrix form.

[1]  J. Lasserre,et al.  Perturbation Analysis of Reduced Resolvents and Generalized Inverses , 2001 .

[2]  Jean B. Lasserre,et al.  The fundamental matrix of singularly perturbed Markov chains , 1999, Advances in Applied Probability.

[3]  Harald K. Wimmer,et al.  The generalized Sylvester equation in polynomial matrices , 1996, IEEE Trans. Autom. Control..

[4]  Alan J. Laub,et al.  An implicit deflation method for ill-conditioned Sylvester and Lyapunov equations , 1995 .

[5]  Bo Kågström,et al.  A Perturbation Analysis of the Generalized Sylvester Equation $( AR - LB,DR - LE ) = ( C,F )$ , 1994, SIAM J. Matrix Anal. Appl..

[6]  V. S. Koroli︠u︡k,et al.  Mathematical Foundations of the State Lumping of Large Systems , 1993 .

[7]  Israel Gohberg,et al.  On the local theory of regular analytic matrix functions , 1993 .

[8]  Nicholas J. Higham,et al.  Perturbation theory and backward error forAX−XB=C , 1993 .

[9]  G. Stewart,et al.  The Laurent expansion of pencils that are singular at the origin , 1991 .

[10]  C. E. de Souza,et al.  Bounds on the solution of the algebraic Riccati equation under perturbations in the coefficients , 1990 .

[11]  A new approach of robust observer design , 1989, Proceedings. ICCON IEEE International Conference on Control and Applications.

[12]  Alan J. Laub,et al.  Controllability and stability radii for companion form systems , 1988, Math. Control. Signals Syst..

[13]  Chia-Chi Tsui,et al.  New approach to robust observer design , 1988 .

[14]  V. G. Gaitsgori,et al.  Theory of Suboptimal Decisions , 1988 .

[15]  Matatyahu Rubin,et al.  The Matrix Equation AX - XB = C , 1984 .

[16]  Chandler Davis,et al.  Perturbation of spectral subspaces and solution of linear operator equations , 1983 .

[17]  B. Moore.  Principal component analysis in linear systems: Controllability, observability, and model reduction , 1981 .

[18]  V. Kučera.  The Matrix Equation $AX + XB = C$ , 1974 .

[19]  M. M. Vaĭnberg,et al.  Theory of branching of solutions of non-linear equations , 1974 .

[20]  P. Lancaster.  Explicit Solutions of Linear Matrix Equations , 1970 .

[21]  Anthony Trampus,et al.  A canonical basis for the matrix transformation X → AXB , 1966 .

[22]  V. Lidskii.  Perturbation theory of non-conjugate operators , 1966 .

[23]  Tosio Kato.  Perturbation theory for linear operators , 1966 .