Research On A Class Of Matrix Equations

Matrix equation theory is a very important part of linear algebra, and it hasimportant applications in algebra, combinatorial, graph theory, control and otherareas. Sylvester matrix equation, which firstly appeared in the1880s, has beenstudied by many researchers.This paper focuses on some of problems about Sylvester matrix equation.Firstly, when the elements of the matrix equation are in the polynomial ring, somenecessary and sufficient conditions for the Sylvester matrix equation and itsgeneralized form has solution is obtained. As a corollary, the problems aboutsimultaneous solutions of systems of matrix equations are studied. Second, thedimension of the solution space of generalized Sylvester matrix equation where theentries belong to the polynomial ring is studied. Finally, the necessary and sufficientcondition of the Sylvester matrix equation and its generalized form which satisfyingthe constraint has solution is obtained. Further, when the constraint condition of thegeneralized Sylvester matrix equation is generalized, the necessary and sufficientcondition for solvability is given.