Differentiable roots, eigenvalues, and eigenvectors

TL;DR

This paper investigates the mathematical conditions under which roots, eigenvalues, and eigenvectors of polynomial and matrix curves are differentiable to a certain degree, providing theoretical insights into their smoothness properties.

Contribution

It establishes the conditions for the existence of differentiable roots, eigenvalues, and eigenvectors for polynomial and matrix curves, advancing understanding of their smoothness.

Findings

Conditions for $C^p$-roots of polynomial curves

Conditions for $C^p$-eigenvalues of matrix curves

Conditions for $C^p$-eigenvectors of matrix curves

Abstract

We determine the conditions for the existence of $C^p$-roots of curves of monic complex polynomials as well as for the existence of $C^p$-eigenvalues and $C^p$-eigenvectors of curves of normal complex matrices.