Linear Preservers of Real Matrix Classes Admitting a Real Logarithm

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.