On Solutions Of Matrix Polynomial Equations

After the introduction of Sylvester equation by Sylvester in1884, Sylvesterequation gradually became a very hot topic in Mathematics field. It has variousapplications in control theory such as pole assignment, robust pole assignment andcharacteristics structure assignment etc. It is of practical importance to investigatethe existence and structure of solutions for this kind of equations.In this thesis, we will firstly give basic definitions and investigate two particulartypes of generalized Sylvester equation utilizing techniques in linear algebra. Wewill also discuss one type of matrix equations above case by case. This type ofequations showing a resemblance to matrix equations satisfied by strong coprimematrix polynomials. After that, we would provide a number of necessary andsufficient conditions for the existence of constant matrix solutions in all cases. Wethen will prove consistency between existence of matrix polynomial solutionsand existence of constant matrix solutions in all other cases aside from a particularcase. Later on, another type of special polynomial matrix equation will beconsidered. We will give equivalent form for equations under some conditions. Wewill also study the set of solution for the considered equation under variousconditions, rank of the free module of solutions in monadic polynomial ring andhow to choose elementary matrices respectively. Finally, we conclude with anexample illustrating the procedures in looking for the homogeneous solutions.