# Eikonal Blog

## 2010.01.26

### Expansions of the exponentials of the sums of matrices

Richmond’s formula #1: $e^{t(A+B)} = e^{tA}e^{tB} + \sum_{r=0}^\infty E_r t^r$, where:

• $E_{r+1} = \frac1{r+1}\left((A+B)E_r+[B,F_r]\right) \$ with $E_0=E_1=0$,
• $F_{r+1} = \frac1{r+1}(A F_r + F_r B) \$ with $F_0=1$.

Richmond’s formula #2: $e^{t(A+B)} = \frac12(e^{tA}e^{tB}+e^{tB}e^{tA}) + \sum_{r=0}^\infty E'_r t^r$, where:

• $E'_{f+1} = \frac1{r+1}((A+B)E'_r+\frac12[B,F_r]+\frac12[A,F^{*}_r]) \$ with $E'_0=E'_1=E'_2=0$, and where $F_r$‘s are the same as in the Richmond’s formula #1.
• Following bounds are valid: $||E'_r|| < \frac1{(r-1)!} (||A||+||B||)^r$.

Note that the exponential generating function ${\cal F}(z):\equiv \sum_{r=0}^\infty \frac{z^r}{r!} F_r$ satisfies the second order ODE: $\partial_z (z\partial_z {\cal F}(z)) = A{\cal F}(z)+{\cal F}(z)B$.

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