# Piecemeal remarks on rational matrix functions of a complex variable

The global theory of rational linear systems on requires the study of (rational) gauge transformations which are holomorphic and holomorphically invertible except for a single point. If this point is at infinity, then the matrix of such transformation is necessarily polynomial with constant nonzero determinant. Such matrix functions are provisionally referred to as *monopoles*, .

Multiplication of a rational matrix function from the left by a monopole matrix corresponds to adding the second row of , multiplied by , to the first row. Thus manipulations with rows of , which aim at Gauss-type elimination of certain monomials from matrix elements, can be represented as gauge actions of the *monopole group*. The principal result that will be used throughout the next few lectures, is the following *Bolibruch Permutation Lemma*.

**Lemma**. Let be the germ of a matrix function, holomorphic and invertible at . Then for any ordered tuple of integer numbers the product , , is monopole equivalent to a product of the form , where is also holomorphic and invertible at , and is a permutation of the tuple .

The proof of this result is not difficult, yet is too technical to be delivered in the classroom.

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