A dense subset of $M\_{n}(\mathbb{R})$ containing diagonalizable matrices

TL;DR

This paper characterizes matrices similar to X-form matrices, proves their set is dense in real and complex matrices, and describes the interior of this set, contributing to understanding matrix similarity and structure.

Contribution

It provides a new characterization of X-form matrices via minimal polynomials and shows their similarity set is dense with a description of its interior.

Findings

The set of matrices similar to X-form matrices is dense in $M_n( eal)$ and $M_n(c)$.

A characterization of the interior of this set is provided.

Matrices similar to X-form matrices include all diagonalizable matrices.

Abstract

In this note, we consider matrices similar to $X$-form matrices, which are the matrices for which only the diagonal and the anti-diagonal elements can be different from zero. First, we give a characterization of these matrices using the minimal polynomial. Then, we prove that the set of matrices similar to $X$-form matrices over $\mathbb{R}$ and $\mathbb{C}$ are dense and we give a characterization of the interior of this set.