The PIMS Postdoctoral Fellow Seminar: Samir Mondal

In real Lie theory, matrices that admit a real logarithm lie in the identity component $\mathrm{GL}_n(\mathbb{R})_+$ of the general linear group $\mathrm{GL}_n(\mathbb{R})$, while their logarithms belong to the Lie algebra $\mathfrak{gl}_n(\mathbb{R})$. The exponential map \[ \exp : M_n(\mathbb{R}) \to \mathrm{GL}_n(\mathbb{R}) \] provides a fundamental link between the Lie algebra and the Lie group, with the logarithm serving as a local inverse.

In this talk, we characterize all bijective linear maps $\varphi : M_n(\mathbb{R}) \to M_n(\mathbb{R})$ that preserve the class of matrices admitting a real (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. Then, using a Zariski density argument, we show that any bijective linear map preserving matrices that admit a real logarithm must also preserve $\mathrm{GL}_n(\mathbb{R})$, which forces the map to be of one of the standard forms.