Eigenvalue paths arising from matrix paths

It is known (see e.g. [2], [4], [5], [6]) that continuous variations in the entries of a complex square matrix induce its eigenvalues. If such variation arises from one real parameter α∈[0,1], then eigenvalues follow paths plane as α shifts 0 to 1. The intent here study nature these eigenpaths, including their behavior under small perturbations variations, well resulting eigenpairings matrices occur at α=0 and α=1. We also give analogs our results setting monic polynomials.