We prove a class of trace inequalities which complements the Golden-Thompson inequality. For example, Tr(epA#epB) 2 p≤Tr eA+B holds for all p>0 when A and B are Hermitian matrices and # denotes the geometric mean. We also prove related trace inequalities involving the logarithmic function; namely p-1TrX log Y p> 2XpY p 2≤Tr X(log X+log Y)≤p-1TrX log X p 2YpX p 2 for all p>0 when X and Y are nonnegative matrices. These inequalities supply lower and upper bounds on the relative entropy.
ASJC Scopus subject areas
- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics