Eikonal Blog

2010.01.28

BCS (Baker–Campbell–Hausdorff) formula

Filed under: Uncategorized — Tags: , — sandokan65 @ 17:12

In this posting the matrix C(t) is defined by e^{C(t)}:\equiv e^{t(A+B)} where A and B are constant matrices.

The Baker–Campbell–Hausdorff theorem claims that:

C(t) = B + \int_0^1  dt g(e^{t a} e^b ) A,

where g(z):\equiv \frac{\ln(z)}{z-1} = \sum_{m=0}^\infty \frac{(1-z)^m}{m+1}, and the lower-case letters a and b represent the adjoint actions of the corresponding matrices A and B (e.g. a X:\equiv ad(A) X :\equiv [A,X]).

One is frequently seeing the following series expression:

C(t) = t (A+B) + \frac{t^2}2 [A,B] + \frac{t^3}{12} ([[A.B],B]-[[A,B],A]) + \cdots = t (A+B) + \frac{t^2}2 a B + \frac{t^3}{12} (b^2 A + a^2 B) + \cdots


Source: T1269 = A.N.Richmond “Expansion for the exponential of a sum of matrices”; Int. Journal of Control (issue and year unknown).

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