Factorizations of Rational Matrix Functions with Applications to Discrete Integrable Systems and Discrete Painleve Equations

Factorizations of Rational Matrix Functions with Applications to Discrete Integrable Systems and Discrete Painlevé Equations

Anton Dzhamay
School of Mathematical Sciences
University of Northern Colorado

We consider the space of rational matrix functions with a given spectral data (i.e., zeroes and poles of the determinant). We are interested in ways of representing matrices from this space as products of elementary blocks and also in determining coordinates on this space that are adapted for such a description. Interchanging the order of these factors results in an interesting integrable discrete dynamical system, together with its Lax representation. We give a Lagrangian description of such systems in the so-called quadratic case, and explain the connection with the isomonodromic transformations of linear systems of difference equations and the difference Painlevé equations.