In physics and mathematics, the Golden–Thompson inequality is a trace inequality between exponentials of symmetric and Hermitian matrices proved independently by Golden (1965) and Thompson (1965).
It has been developed in the context of statistical mechanics, where it has come to have a particular significance.
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 the right hand side of the inequality, this can be seen by rewriting it as
using the cyclic property of the trace.
denote the Frobenius norm, then the Golden–Thompson inequality is equivalently stated as
The Golden–Thompson inequality can be viewed as a generalization of a stronger statement for real numbers.
This relationship is not true if A and B do not commute.
In fact, Petz (1994) 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.
are not equal, they are still related by an inequality.
are Hermitian and positive semidefinite, then
, then all the other inequalities are also proven as special cases of it.
So it suffices to prove that inequality.
are Hermitian and PSD, we can split
, meaning it is a non-negative real number.
Define two sequences of matrices
which, by construction, are Hermitian and positive semidefinite.
Golden–Thompson inequality (Thompson (1965)) — Given Hermitian matrices
In general, if A and B are Hermitian matrices and
is a unitarily invariant norm, then (Bhatia 1997, Theorem IX.3.7) The standard Golden–Thompson inequality is a special case of the above inequality, where the norm is the Frobenius norm.
The general case is provable in the same way, since unitarily invariant norms also satisfy the Cauchy-Schwarz inequality.
(Bhatia 1997, Exercise IV.2.7) Indeed, for a slightly more general case, essentially the same proof applies.
limit, we obtain the operator norm
The second claim is proven similarly.
The inequality has been generalized to three matrices by Lieb (1973) and furthermore to any arbitrary number of Hermitian matrices by Sutter, Berta & Tomamichel (2016).
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
is the derivative of the matrix logarithm given by
, and the inequality for three matrices reduces to the original from Golden and Thompson.
Bertram Kostant (1973) used the Kostant convexity theorem to generalize the Golden–Thompson inequality to all compact Lie groups.