Short proofs of theorems of Mirsky and Horn on diagonals and eigenvalues of matrices

TL;DR

This paper presents a simple inductive proof of Mirsky's theorem, which characterizes when a complex matrix can have specified diagonals and eigenvalues, advancing understanding of matrix spectral properties.

Contribution

The paper offers a straightforward inductive proof of Mirsky's theorem, simplifying the existing proof and deepening insight into matrix diagonal and eigenvalue relationships.

Findings

Provided a simple inductive proof of Mirsky's theorem.

Clarified conditions for matrix diagonals and eigenvalues.

Enhanced understanding of matrix spectral properties.

Abstract

A theorem of Mirsky provides necessary and sufficient conditions for the existence of an N-square complex matrix with prescribed diagonal entries and prescribed eigenvalues. We give a simple inductive proof of this theorem.