Golden–Thompson inequality
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: 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 formula_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+b" is the product of the exponential of "a" with the exponential of "b": If we replace "a" and "b" with commuting matrices "A" and "B", then the same inequality formula_4 holds. This relationship is not true if "A" and "B" do not commute. In fact, proved that if "A" and "B" are two Hermitian matrices for which the Golden–Thompson inequality is  verified as an equality, then the two matrices commute. The Golden–Thompson inequality shows that, even though formula_5 and formula_6 are not equal, they are still related by an inequality. Generalizations. The Golden–Thompson inequality generalizes to any unitarily invariant norm. If "A" and "B" are Hermitian matrices and formula_7 is a unitarily invariant norm, then The standard Golden–Thompson inequality is a special case of the above inequality, where the norm is the Schatten norm with formula_9. Since formula_5 and formula_11 are both positive semidefinite matrices, formula_12 and formula_13. The inequality has been generalized to three matrices by and  furthermore to any arbitrary number of Hermitian matrices by . A naive attempt at generalization does not work: the inequality is false. For three matrices, the correct generalization takes the following form: where the operator formula_16 is the derivative of the matrix logarithm given by formula_17. Note that, if formula_18 and formula_19 commute, then formula_20, and the inequality for three matrices reduces to the original from Golden and Thompson.