Research On Problems Related To Sylvester Matrix Equations

Matrix equations are very important in many fields.Matrix equations are ubiquitous in systems theory,automatic control,stability theory,optimization theory and the like.This article will focus on problems related to Sylvester matrix equations.Firstly,we discuss the solvability of T-Sylvester matrix equation over rings with 2being invertible.By transforming the problem on commutative rings to Artin rings,we construct two modules and establish the map between them for analyzing.We generalize the theorem given by Wimmer about necessary and sufficient conditions for the solvability of matrix equation on the complex field.Secondly,we study the solvability and the property of solutions of generalized Sylvester matrix equation on the polynomial ring over a field F.By matrix polynomial division with remainder,we just need to focus on the matrix equation problem in the low-order situation.By constructing a new equation,the study of T-Sylvester matrix equation on the polynomial ring F[?] can be reduced to study Sylvester matrix equation on F.By using the technology of tensor product of matrices,we study necessary and sufficient conditions for the solvability of Sylvester matrix equation,so that the number of the parameters of the new equation is less.Thirdly,we discuss singular vectors that correspond to the largest singular value of the generalized Lyapunov operator.We construct a counter-example to note on the mistake of the proof of that singular vectors corresponding to largest singular value of classic Lyapunov operator are symmetric matrix.Thus it is still a conjecture.We prove that the conjecture is correct if the order of the matrix is less than 6.Furthermore,we discuss in detail the singular vectors of generalized continuous and discrete Lyapunov operator.We prove that the singular vectors of generalized Lyapunov operator that correspond to the largest singular value are symmetric matrix when the order of the matrix less than 4.When the order of matrix are greater than or equal to 4,we give an example to illustrate that the conjecture is not always true.Finally,we also discuss the singular vectors corresponding to the minimum singular value of generalized Lyapunov operator.If the order of the matrix is less than 3,the singular vectors corresponding to the minimum singular value can always be symmetric matrix.If the order is greater than 2,we give counter examples which shows that singular vectors are not always symmetric.We prove that,for the stable matrix pencil,the singular vectors corresponding minimum singular value of generalized Lyapunov operator can always be taken as a symmetric matrix.