The linear preservers of real diagonalizable matrices

TL;DR

This paper characterizes all linear transformations of real matrices that preserve diagonalizability, building on recent results, and explores the structure of maximal diagonalizable subspaces.

Contribution

It provides a complete description of automorphisms stabilizing diagonalizable matrices, extending the understanding of linear preservers in matrix theory.

Findings

Identifies all automorphisms of M_n(R) that preserve diagonalizability.

Analyzes the structure of maximal diagonalizable subspaces.

Builds on recent results by Bogdanov and Guterman.

Abstract

Using a recent result of Bogdanov and Guterman on the linear preservers of pairs of simultaneously diagonalizable matrices, we determine all the automorphisms of the vector space M_n(R) which stabilize the set of diagonalizable matrices. To do so, we investigate the structure of linear subspaces of diagonalizable matrices of M_n(R) with maximal dimension.