Trace and determinant preserving maps of matrices

2016/03/12 by Huang, Huajun, Liu, Chih-Neng, Szokol, Patricia +2 · 1 citation
#15A04 #15A15 #FOS: Mathematics #Functional Analysis (math.FA) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1603.03869

Abstract

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

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