In this chapter we discuss both the general theory of quadratic forms, the notion of congruence and its invariants, and the applications of the theory to the analysis of special forms. We focus on quadratic forms induced by rational functions, most notably the Hankel and Bezout forms, because of their connection to system-theoretic problems like stability and signature-symmetric realizations. These forms use as their data different representations of rational functions, power series, and coprime factorizations, respectively. But we also will discuss the partial fraction representation in relation to the computation of the Cauchy index of a rational function, the proof of the Hermite-Hurwitz theorem, and the continued fraction representation as a tool in the computation of signatures of Hankel matrices as well as in the problem of Hankel matrix inversion. Thus, different representations of rational functions, that is, different encodings of the information carried by a rational function, provide efficient starting points for different methods. The results obtained for rational functions are applied to root location problems for polynomials in the next chapter.
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