## Introduction

In order to introduce the whole family of Sylvester matrix equations, three types of dynamical models for linear systems need to be first mentioned.

1.1 Three Types of Linear Models 1.1.1 First-Order Systems In certain applications, we often encounter a type of first-order linear systems, which are represented by the following model:

{ ExË™ âˆ’ Ax = B1uË™ + B0u y = Cx + Du, (1.1)

where

â– x âˆˆ Rq, y âˆˆ Rm, and u âˆˆ Rr are the state vector, output vector, and the control vector, respectively

â– E , A âˆˆ RnÃ—q, B1,B0 âˆˆ RnÃ—r , C âˆˆ RmÃ—q, and D âˆˆ RmÃ—r are the system coefficient matrices

This is a general first-order rectangular system (Duan 2010, Fletcher 1997, Habets et al. 2006, Hou 2004, Ishihara and Terra 2001, Zhang 2006). Although rectangular systems are often encountered, in most applications, we deal with square systems, that is, systems in the above form, but with q = n.