Golden–Thompson inequality

In physics and mathematics, the Golden–Thompson inequality, proved independently by Golden (1965) and Thompson (1965), says that for Hermitian matrices A and B,

$\operatorname{tr}\, e^{A+B} \le \operatorname{tr} \left(e^A e^B\right)$

where tr is the trace, and e^A is the matrix exponential. This trace inequality is of particular significance in statistical mechanics, and was first derived in that context.

Bertram Kostant (1973) used the Kostant convexity theorem to generalize the Golden–Thompson inequality to all compact Lie groups.

References

• J.E. Cohen, S. Friedland, T. Kato, F. Kelly, Eigenvalue inequalities for products of matrix exponentials, Linear algebra and its applications, Vol. 45, pp. 55–95, 1982. doi:10.1016/0024-3795(82)90211-7
• D. Petz, A survey of trace inequalities, in Functional Analysis and Operator Theory, 287–298, Banach Center Publications, 30 (Warszawa 1994).