Lie product formula
In mathematics, the Lie product formula, named for Sophus Lie (1875), but also widely called the Trotter product formula, named after Hale Trotter, states that for arbitrary m × m real or complex matrices A and B, e A + B = lim n → ∞ ( e A / n e B / n ) n , {\displaystyle e^{A+B}=\lim _{n\rightarrow \infty }(e^{A/n}e^{B/n})^{n},} where eA denotes the matrix exponential of A. The Lie–Trotter product formula and the Trotter–Kato theorem extend this to certain unbounded linear operators A and B. This formula is an analogue of the classical exponential law e x + y = e x e y {\displaystyle e^{x+y}=e^{x}e^{y}} which holds for all real or complex numbers x and y. If x and y are replaced with matrices A and B, and the exponential replaced with a matrix exponential, it is usually necessary for A and B to commute for the law to still hold.