Exponential splittings « Eikonal Blog

2010.01.26

A Theorem of Ellis and Pinsky

Filed under: mathematics — Tags: Exponential splittings — sandokan65 @ 13:12

Theorem (Ellis & Pinsky): Let $\latex A$, $B$ be real symmetric $n\times n$ matrices, and assume that $B$ is negative semi-definite. Then as $\epsilon \rightarrow +0$ :

$e^{t(A+i\frac1{\epsilon}B)} e^{-i\frac{t}{\epsilon}B} = e^{t A_0} + {\cal O}(\epsilon)$ ,

where $A_0 :\equiv \sum_{\lambda\in\sigma(B)} P_\lambda A P_\lambda$ , here $P_\lambda$ being the orthogonal projector on the eigenspace of $B$ associated with the eigenvalue $\lambda$ .

Source: T1270 = S.L.Campbell “On the limit of a product of matrix exponentials”; Linear and Multilinear Algebra; 1978, Vol 6, p.p. 55-59.

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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).

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