Inverse Spectral Problems for Regular Improper Rational Matrix Functions

We consider the problem of constructing a regular rational n × n matrix function W(z) = C(zI - A) -1B + D + zE(I - zG)-1F such that WR n + = W1R n + and WR n - = W2R n - . Here R n + (respectively R n - ) is the space of rational ℂn-valued functions analytic inside (respectively outside) a smooth closed con our in the complex plane, and realizations Wj(z) = Cj (zI - Aj)-1Bj + Dj + zEj (I - zGj)-1Fj are given for j = 1, 2. The special case where the contour Γ is the unit circle and W1 (∞) = W2 (∞) = I was studied recently in [BR3]. As applications we consider the model reduction problem from linear systems theory for both the discrete time and continuous time settings and the special case of matrix polynomials.

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