The solution of the Loewy-Radwan conjecture

TL;DR

This paper proves the Loewy-Radwan conjecture by determining the maximal dimension and structure of linear spaces of matrices with limited eigenvalues, extending classical results on nilpotent matrices.

Contribution

It provides a positive solution to the Loewy-Radwan conjecture, establishing the maximal dimension and structure of matrix spaces with bounded eigenvalues.

Findings

Maximal dimension of matrix spaces with at most k eigenvalues

Structural characterization of extremal matrix spaces

Extension of Gerstenhaber's results to broader eigenvalue constraints

Abstract

A seminal result of Gerstenhaber gives the maximal dimension of a linear space of nilpotent matrices. It also exhibits the structure of this space where the maximal dimension is attained. Extensions of this result in the direction of linear spaces of matrices with a bounded number of eigenvalues have been studied. In this paper, we answer perhaps the most general problem of the kind as proposed by Loewy and Radwan by solving their conjecture in the positive. We give the dimension of a maximal linear space of $n\times n$ matrices with no more than $k<n$ eigenvalues. We also exhibit the structure of the space where this dimension is attained.