New tools in the matrix fraction description of linear systems

The paper reviews the definition of the generalized Bezoutian matrix, which is defined using the coefficients of the matrix polynomials in prescribed left and right matrix fraction decompositions of a rational transfer function matrix W(s). Like its classical counterpart the generalized Bezoutian matrix is relevant for studying a wide variety of problems. For example, its rank gives information about the minimality or otherwise of the matrix fraction decompositions and the degree of the rational transfer function matrix. In case W(s) is symmetric, the generalized Bezoutian matrix can be used to define a matrix Cauchy index. In case W(s) is lossless, RC or RL positive real, the generalized Bezoutian matrix possesses certain properties. Another use of the Bezoutian is for the characterization of square nonsignular polynomial matrices which have stable determinants.