Linear Preservers of Real Matrix Classes Admitting a Real Logarithm

TL;DR

This paper characterizes all bijective linear maps on real matrices that preserve the class of matrices admitting a real logarithm, showing they are essentially conjugations or transpose conjugations scaled by a positive factor.

Contribution

It provides a complete description of linear preservers of matrices with real logarithms, extending understanding of structure-preserving maps in Lie theory.

Findings

Preservers are of the form conjugation or transpose conjugation scaled by a positive constant.

Any bijective linear map preserving matrices with real logarithms must preserve the general linear group.

The proof uses analysis within standard transformations and Zariski denseness arguments.

Abstract

In real Lie theory, matrices that admit a real logarithm reside in the identity component $\mathrm{GL}_n(\mathbb{R})_+$ of the general linear group $\mathrm{GL}_n(\mathbb{R})$, with logarithms in the Lie algebra $\mathfrak{gl}_n(\mathbb{R})$. The exponential map \[ \exp : \mnr \to \mathrm{GL}_n(\mathbb{R}) \] provides a fundamental link between the Lie algebra and the Lie group, with the logarithm as its local inverse. In this paper, we characterize all bijective linear maps $\varphi : \mnr \to \mnr$ that preserve the class of matrices admitting a real logarithm (principal logarithm). We show that such maps are exactly those of the form \[ \varphi(A) = c\, P A P^{-1} \quad \text{or} \quad \varphi(A) = c\, P A^{T} P^{-1}, \] for some $P \in \mathrm{GL}_n(\mathbb{R})$ and $c > 0$. The proof proceeds in two stages. First, we analyze preservers within the class of standard linear transformations. Second, using Zariski denseness, we prove that any bijective linear map preserving matrices with real logarithms (principal logarithm) must preserve $\mathrm{GL}_n(\mathbb{R})$, which then implies the map is of the standard form.