Homotopic deviation theory for regular matrix pencils

Abstract

We generalize the theory of homotopic deviation of square complex matrices to regular matrix pencils. To this end, we study the existence and the analyticity of the resolvent of the matrix pencils whose matrices are under homotopic deviation with the deviation parameter . Moreover, we investigate and identify the limits of both the resolvent and the spectrum of the deviated matrix pencils, as . We also study the special cases where t tends to the eigenvalues of the related matrix pairs. We use the notions and the results of the generalized homotopic deviation theory to analyze the Weierstrass structure of the deviated matrix pencils under two different assumptions, in particular, either the eigenvalues of the deviated matrix pencils are independent parameters, or the deviation parameter t is an independent parameter. Numerical examples illustrate and support the theoretical results.