##### Abstract

In this thesis the right and left pole structure of a not necessarily regular rational matrix function W is described in terms of pairs of matrices-right and left pole pairs. The concept of orthogonality in R^{n} is investigated. Using this concept, the right and left zero structure of a rational matrix function W is described in terms of pairs and triples of matrices-right and left null pairs and right and left kernel triples. The definition of a spectral triple of a regular rational matrix function over a subset σ of C is extended to the nonregular case. Given a rational matrix function W and a subset σ of C, the left null-pole subspace of W over σ is described in terms of a left kernel triple and a left σ-spectral triple for W. A sufficient condition for the minimality of McMillan degree of a rational matrix function H which is right equivalent to W on σ, that is a rational matrix function H of the same size and with the same left null-pole subspace over σ as W, is developed. An algorithm for constructing a rational matrix function W with a left kernel triple (A_{κ} B_{κ} D_{κ}) and left null and right pole pairs over σ⊂C (A_{ζ}, B_{ζ}) and (C_{π}, A_{π}), respectively, from a regular rational matrix function with left null and right pole pairs over σ (A_{ζ}, B_{ζ}) and (C_{π}, A_{π}) is described. Finally, a necessary and sufficient condition for existence of a rational matrix function W with a given left kernel triple and a given left spectral triple over a subset σ of C is established.