The space of all p-th roots of a nilpotent complex matrix is path-connected

TL;DR

This paper proves that for any nilpotent complex matrix, the set of all its p-th roots forms a path-connected space, revealing a topological property of matrix roots.

Contribution

The paper establishes the path-connectedness of the set of p-th roots of nilpotent matrices, a new topological insight in matrix theory.

Findings

The set of p-th roots of a nilpotent matrix is path-connected.

Path-connectedness holds for all positive integers p.

The result applies to complex nilpotent matrices.

Abstract

Let p be a positive integer and A be a nilpotent complex matrix. We prove that the set of all p-th roots of A is path-connected.