Concept

# Golden–Thompson inequality

Summary
In physics and mathematics, the Golden–Thompson inequality is a trace inequality between exponentials of symmetric and Hermitian matrices proved independently by and . It has been developed in the context of statistical mechanics, where it has come to have a particular significance. Statement The Golden–Thompson inequality states that for (real) symmetric or (complex) Hermitian matrices A and B, the following trace inequality holds: : \operatorname{tr}, e^{A+B} \le \operatorname{tr} \left(e^A e^B\right). This inequality is well defined, since the quantities on either side are real numbers. For the expression on right hand side of the inequality, this can be seen by rewriting it as \operatorname{tr}(e^{A/2}e^B e^{A/2}) using the cyclic property of the trace. Motivation The Golden–Thompson inequality can be viewed as a generalization of a stronger statement for real numbers. If a and b are two real numbers, then the exponential of a+
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