Trace and determinant preserving maps of matrices

TL;DR

This paper characterizes maps on matrices that preserve determinants and traces, showing they are essentially conjugations or transpositions by an invertible matrix, with applications to various matrix classes.

Contribution

It provides a complete characterization of trace and determinant preserving maps on matrices, extending to different matrix sets and revealing their structural form.

Findings

Maps preserving determinant of sums induce trace-preserving properties.

Such maps are of the form $\,A o M^*AM$ or $A o M^*A^tM$.

The results apply to complex, positive definite, symmetric, and upper triangular matrices.

Abstract

Suppose a map $\phi$ on the set of positive definite matrices satisfies $\det(A+B)=\det(\phi(A)+\phi(B))$. Then we have $${\rm tr}(AB^{-1}) = {\rm tr}(\phi(A){\phi(B)}^{-1}).$$ Through this viewpoint, we show that $\phi$ is of the form $\phi(A)= M^*AM$ or $\phi(A)= M^*A^tM$ for some invertible matrix $M$ with $\det (M^*M)=1$. We also characterize the map $\phi: \mathcal{S} \rightarrow \mathcal{S}$ preserving the determinant of convex combinations in $\mathcal{S}$ by using similar method. Here $\mathcal{S}$ can be the set of complex matrices, positive definite matrices, symmetric matrices, and upper triangular matrices.