Nonsingular systems of generalized Sylvester equations: An algorithmic approach

Abstract

Summary We consider the uniqueness of solution (i.e., nonsingularity) of systems of r generalized Sylvester and ⋆‐Sylvester equations with n×n coefficients. After several reductions, we show that it is sufficient to analyze periodic systems having, at most, one generalized ⋆‐Sylvester equation. We provide characterizations for the nonsingularity in terms of spectral properties of either matrix pencils or formal matrix products, both constructed from the coefficients of the system. The proposed approach uses the periodic Schur decomposition and leads to a backward stable O(n3r) algorithm for computing the (unique) solution.

Palindromic linearization and numerical solution of nonsymmetric algebraic T-Riccati equations

“…By post-multiplying the latter equation by D T −B T X and by using (1), one gets (9). The case where det(D T − B T X) = 0 can be treated analogously.…”

“…The numerical solution of ( 22) can be carried out by, e.g., [9,Algorithm 2]. In our numerical experiments, since the matrix variable are just vectors, we solve the linear system of equations arising by Kronecker transformations, namely…”

“…Indeed, recently, there has been a great interest in the T counterpart of classical linear matrix equations. For instance, look at the Sylvester equation AX + XB = C, whose T counterpart AX T + XB = C has been the subject of a number of both computational and theoretical papers; see, e.g., [5], [6], [7], [8], [9], [10], [15]. It is well-known that there is a strict connection between equation ( 2) and the matrix…”

Functional observer design using the direct method for discrete time-varying linear systems

“…However, up to now, there is no general method available to effectively solve the Sylvester equation, and every approach proposes conditions on the existence and uniqueness of the GSE solution. 14,15 The method based on the parametric solution of the GSE [16][17][18][19][20] has the advantage to provide all the design degrees of freedom, so that additional performances in control system design can be optimised. However, this method requires that the observer eigenvalues are distinct and stable but not fixed in advance.…”

On the Low-Degree Solution of the Sylvester Matrix Polynomial Equation

“…where the matrices involved are the matrix polynomials (e.g., [1][2][3][4][5][6][7]). ere have been an extensive study and application of the generalized Sylvester matrix equation (e.g., [8][9][10][11]). is work investigates the bound of the low-degree solution of equation ( 1) with degree 1 matrix polynomials A and B.…”