Generalization of Mirsky's theorem on diagonals and eigenvalues of matrices

TL;DR

This paper generalizes Mirsky's theorem to matrices over any field, providing a short proof and establishing the uniqueness of a companion-matrix-type solution for given eigenvalues and diagonals.

Contribution

The paper extends Mirsky's theorem to arbitrary fields and demonstrates the uniqueness of the solution in a companion-matrix form.

Findings

Generalization of Mirsky's theorem to any field.

Proof of the theorem's sufficiency condition.

Uniqueness of the companion-matrix-type solution.

Abstract

Mirsky proved that, for the existence of a complex matrix with given eigenvalues and diagonal entries, the obvious necessary condition is also sufficient. We generalize this theorem to matrices over any field and provide a short proof. Moreover, we show that there is a unique companion-matrix-type solution for this problem.