A note on Sylvester-type equations

This work is to provide a comprehensive treatment of the relationship between the theory of the generalized (palindromic) eigenvalue problem and the theory of the Sylvester-type equations. Under a regularity assumption for a specific matrix pencil, we show that the solution of the ⋆-Sylvester matrix equation is uniquely determined and can be obtained by considering its corresponding deflating subspace. We also propose an iterative method with quadratic convergence to compute the stabilizing solution of the ⋆-Sylvester matrix equation via the well-developed palindromic doubling algorithm. We believe that our discussion is the first which implements the tactic of the deflating subspace for solving Sylvester equations and could give rise to the possibility of developing an advanced and effective solver for different types of matrix equations.

[1]  Masoud Hajarian,et al.  Matrix iterative methods for solving the Sylvester-transpose and periodic Sylvester matrix equations , 2013, J. Frankl. Inst..

[2]  Guoliang Chen,et al.  Iterative methods for solving linear matrix equation and linear matrix system , 2010, Int. J. Comput. Math..

[3]  Daniel Kressner,et al.  Implicit QR algorithms for palindromic and even eigenvalue problems , 2009, Numerical Algorithms.

[4]  E. Chu,et al.  Vibration of fast trains, palindromic eigenvalue problems and structure-preserving doubling algorithms , 2008 .

[5]  V. Mehrmann The Autonomous Linear Quadratic Control Problem , 1991 .

[6]  E. Chu,et al.  PALINDROMIC EIGENVALUE PROBLEMS: A BRIEF SURVEY , 2010 .

[7]  Harald K. Wimmer Roth's theorems for matrix equations with symmetry constraints , 1994 .

[8]  Leiba Rodman,et al.  Algebraic Riccati equations , 1995 .

[9]  G. W. Stewart,et al.  Computer Science and Scientific Computing , 1990 .

[10]  Ilse C. F. Ipsen Accurate Eigenvalues for Fast Trains , 2004 .

[11]  Wen-Wei Lin,et al.  A structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equation , 2006, Numerische Mathematik.

[12]  V. Mehrmann The Autonomous Linear Quadratic Control Problem: Theory and Numerical Solution , 1991 .

[13]  Daniel Kressner,et al.  Structured Condition Numbers for Invariant Subspaces , 2006, SIAM J. Matrix Anal. Appl..

[14]  P. Lancaster,et al.  Factorization of selfadjoint matrix polynomials with constant signature , 1982 .

[15]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[16]  Volker Mehrmann,et al.  Numerical methods for palindromic eigenvalue problems: Computing the anti‐triangular Schur form , 2009, Numer. Linear Algebra Appl..

[17]  Volker Mehrmann,et al.  Vector Spaces of Linearizations for Matrix Polynomials , 2006, SIAM J. Matrix Anal. Appl..

[18]  G. Stewart,et al.  Matrix Perturbation Theory , 1990 .

[19]  Tiexiang Li,et al.  The palindromic generalized eigenvalue problem A∗x=λAx: Numerical solution and applications , 2011 .

[20]  K. Weierstrass Zur Theorie der bilinearen und quadratischen Formen , 2013 .

[21]  Wen-Wei Lin,et al.  On the ⋆-Sylvester equation AX ± X⋆ B⋆ = C , 2012, Appl. Math. Comput..

[22]  Fernando De Ter,et al.  CONSISTENCY AND EFFICIENT SOLUTION OF THE SYLVESTER EQUATION FOR ⋆-CONGRUENCE , 2011 .

[23]  J. Doyle,et al.  Robust and optimal control , 1995, Proceedings of 35th IEEE Conference on Decision and Control.

[24]  Lv Tong,et al.  The solution to Matrix Equation , 2002 .

[25]  Volker Mehrmann,et al.  ON THE SOLUTION OF PALINDROMIC EIGENVALUE PROBLEMS , 2004 .

[26]  Volker Mehrmann,et al.  Structured Polynomial Eigenvalue Problems: Good Vibrations from Good Linearizations , 2006, SIAM J. Matrix Anal. Appl..

[27]  Qingling Zhang,et al.  The solution to matrix equation AX+XTC=B , 2007, J. Frankl. Inst..

[28]  K. Chu The solution of the matrix equations AXB−CXD=E AND (YA−DZ,YC−BZ)=(E,F) , 1987 .