The canonical structure of a pencil of degenerate matrix functions

In this paper we study global properties of pencil of identically degenerate matrix functions with a compact domain of definition. Matrix functions are assumed to have a constant rank and all roots of the characteristic equation of matrix pencil are assumed to have a constant multiplicity at each point in the domain of definition. We obtain sufficient conditions for the smooth orthogonal similarity of matrix functions to the upper triangular form and sufficient conditions for the smooth equivalence of the pencil of matrix functions to its canonical form. We illustrate the obtained results with simple examples.