# The Golden-Thompson inequality

Let ${A, B}$ be two Hermitian ${n \times n}$ matrices. When ${A}$ and ${B}$ commute, we have the identity

$\displaystyle e^{A+B} = e^A e^B.$

When ${A}$ and ${B}$ do not commute, the situation is more complicated; we have the Baker-Campbell-Hausdorff formula

$\displaystyle e^{A+B} = e^A e^B e^{-\frac{1}{2}[A,B]} \ldots$