Rational Matrix Functions

Abstract

This chapter contains a specification in detailed form of the results of Chapter 3 for rational matrix functions, and the solution of the next interpolation problem, namely to build a rational matrix function with a given null and pole structure. Unlike the scalar case, it turns out that even when the solution exists and the value at infinity is specified, it may not be unique. A particular solution can be singled out if additional information is provided. This extra information consists of an invertible matrix, called the coupling matrix, which describes the geometry of the interaction between pole and null structures at each point. If a solution exists and the poles and zeros are disjoint, then, as in the scalar case, the solution is unique. The solution of the interpolation problem is obtained in a special form called a realization for the matrix function. The theory of realization for a rational matrix function is borrowed from systems theory; a short exposition of the necessary facts concerning realization is also presented in the chapter. In this chapter we consider only the case where the rational matrix function is analytic and invertible at infinity, and the interpolation data consist of a left null pair together with a right pole pair. This combination (as well as the reversed combination of right null and left pole pair) we call standard. (Nonstandard combinations will be considered in Chapter 8.) The solution of the interpolation problem considered in this chapter, if the value at infinity is not fixed, may be multiplied on the left by any invertible constant matrix to generate another solution of the same problem. Thus we see that solvable interpolation problems associated with standard local data are matricially homogeneous.