where X ∈ M(n,m)(K), play a central role in many areas of applied mathematics and in particular in systems and control theory. It is well known that if K is an algebraically closed field then the matrix equation (1) possesses a unique solution if and only if the matrices A and B have no common eigenvalues (see [[3]] and [11]). In this work we give a brief survey of methods used to solve the (SM...

For when the Sylvester matrix equation has a unique solution, this work provides a closed form solution, which is expressed as a polynomial of known matrices. In the case of non-uniqueness, the solution set of the Sylvester matrix equation is a subset of that of a deduced equation, which is a system of linear algebraic equations. c © 2005 Elsevier Ltd. All rights reserved.

The matrix Sylvester equation for congruence, or T-Sylvester equation, has recently attracted considerable attention as a consequence of its close relation to palindromic eigenvalue problems. The theory concerning T-Sylvester equations is rather well understood and there are stable and efficient numerical algorithms which solve these equations for smallto medium-sized matrices. However, develop...

The matrix Sylvester equation for congruence, or T-Sylvester equation, has recently attracted considerable attention as a consequence of its close relation to palindromic eigenvalue problems. The theory concerning T-Sylvester equations is rather well understood and there are stable and efficient numerical algorithms which solve these equations for smallto medium-sized matrices. However, develop...

We provide necessary and sufficient conditions for the generalized ⋆Sylvester matrix equation, AXB+CX ⋆ D = E, to have exactly one solution for any right-hand side E. These conditions are given for arbitrary coefficient matrices A,B,C,D (either square or rectangular) and generalize existing results for the same equation with square coefficients. We also review the known results regarding the ex...

We provide necessary and sufficient conditions for the generalized ?Sylvester matrix equation, AXB +CX ? D = E, to have exactly one solution for any right-hand side E. These conditions are given for arbitrary coefficient matrices A,B,C,D (either square or rectangular) and generalize existing results for the same equation with square coefficients. We also review the known results regarding the e...

An efficient computational method is presented for state space analysis of singular systems via Haar wavelets. Singular systems are those in which dynamics are governed by a combination of algebraic and differential equations. The corresponding differential-algebraic matrix equation is converted to a generalized Sylvester matrix equation by using Haar wavelet basis. First, an explicit expressio...

This paper studies a class of real-rational matrix bilateral Diophantine equations (BDE) arising in numerous control problems. A necessary and sufficient solvability condition is derived in terms of state-space realizations of rational matrices involved in the equation. This condition is given in terms of a constrained matrix Sylvester equation and is numerically tractable. An explicit state-sp...

In this paper, an explicit, analytical and complete solution to the generalized discrete Sylvester matrix equation M X N − X = T Y which is closely related with several types of matrix equations in control theory is obtained. The proposed solution has a neat and elegant form in terms of the Krylov matrix, a block Hankel matrix and an observability matrix. Based on the proposed solution, an expl...

The nonlinear matrix equation X+A∗X−1A = Q can be cast as a linear Sylvester equation subject to unitary constraint. The Sylvester equation can be obtained by means of hermitian eigenvalue computation. The unitary constraint can be satisfied by means of either a straightforward alternating projection method or by a coordinate-free Newton iteration. The idea proposed in this paper originates fro...