Golden–Thompson inequality

For Hermitian matrices A and B, the Golden–Thompson inequality states that tr(e^(A+B)) ≤ tr(e^A e^B). It bounds a matrix exponential of a sum using the product of matrix exponentials.

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Matrix exponential: The exponential of a square matrix, defined by its power series or spectral calculus. Exponentials of A, B, and their sum are the quantities compared.
Beppo Levi: An Italian mathematician known for work in analysis, measure theory, and mathematical physics. Levi published an early proof of the trace inequality in 1937.
Matrix concentration inequality: A probabilistic bound on deviations of sums of random matrices from their expectations. The inequality helps derive tail bounds for eigenvalues of random matrix sums.
Löwner order: The partial order on Hermitian matrices defined by positive semidefinite differences. Golden–Thompson compares traces; it does not generally assert e^(A+B) ≤ e^A e^B in this order.
Matrix trace: The sum of the diagonal entries of a square matrix, invariant under change of basis. Taking the trace turns the operator comparison into a scalar inequality.
Theodore W. Anderson: An American statistician whose research included probability, matrix inequalities, and statistical theory. Anderson independently established the inequality in 1960.
Matrix Chernoff bound: A tail bound for extreme eigenvalues of sums of independent positive semidefinite random matrices. Its proofs use trace-exponential estimates built from Golden–Thompson.
Peierls–Bogoliubov inequality: A variational inequality relating free energies of a quantum system and a trial state. It is another trace-based bound used in quantum statistical mechanics, with different hypotheses and purpose.
Noncommutative algebra: Algebraic systems in which multiplication need not be commutative. The inequality is useful precisely when A and B need not commute.
Hiroshi H. H. Chan: A mathematician whose work includes matrix inequalities and operator theory. Chan independently proved the result in 1960, completing the theorem’s familiar name.