A Functional Method For Bezoutians Over An Arbitrary Field

In this paper we study the polynomial Bezoutians over an arbitrary algebra field F bymeans of functional methods, and then the minimal Hermitian symmetric realizations ofcompound rational matrix functions are introduced, which have important applications inthe theory of linear control systems.In chapter one we introduce the history and present situation of Bezoutian, and thepaper’s main work is listed.In chapter two we study properties of polynomial Bezoutians over an arbitrary fieldby means of functional methods, such as Bezoutian diagonalization, Barnett factorization,stability of Bezoutian and so on, then the close relations between Bezoutian and triplerealization of linear control systems are showed by Barnett factorization and stability ofBezoutian.In chapter three we prove that the product of companion matrix C and Bezoutian B(x,y) of polynomials f(x) and g(x) is also a Bezoutian by Vandermonde diagonalization ofcompanion matrix C of polynomial f(x) over the algebra field F，then the finite linearcombinations of product of polynomial Bezoutians and arbitrary nonnegative integer powerof companion matrix C also have the similar property. Especially, if the matrix C is fullrank, the negative integer power of matrix C also have the same property, and we give thegenerating functions of these two types of polynomial Bezoutians. These two types ofBezoutians are called Bezoutian pencil of polynomial f(x) and companion matrix Crespectively.In chapter four the theory of minimal Hermitian realizations of compound rationalmatrix functions is put forward. We give a minimal realizations of compound rationalmatrix functionsf (W (λ))for a regular Hermitian matrixW (λ)on the real axis and apolynomial functionf (λ)over the field of real numbers. Moreover, we prove that theminimal realizations off (W (λ))can be symmetrized by the symmetrization method ofminimal realizations of rational Hermitian matrix functions in reference [37].