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Let $X$ be an $n\times n$ matrix, $u,v$ are two vectors. Can we express $e^{X+uv^T}$ in terms of $e^X$ and $e^{uv^T}$? Is there a concise formula? I know there is a Lie product formula http://en.wikipedia.org/wiki/Lie_product_formula, but it depends on a limit.

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    If $i \neq j$ then, writing $Y = u v^T$, we have $[Y,[Y,Z]]=0$ for any $Z$. This should make Baker-Campbell-Hausdorff simplify a lot. I suspect there are also simplifications available when $i = j$, but I don't see what they are right now.2011-11-01

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I don't know if this directly helps, but computationally you could do the following:

  1. Obtain a rational approximation to exp(X)
  2. Use the following software: http://www.cs.cornell.edu/cv/ResearchPDF/GenSherMorr.htm which computes rank-1 updates to rational functions of matrices.

Naturally, for specialized choices of $X$, $u$, $v$, one can obtain better algorithms.

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    "Obtain a rational approxi$m$ation" - the Padé approxi$m$ants are particularly convenient. [Here](http://dx.doi.org/10.1137/S0895479898333636) is the paper discussing the generalization of the Sherman-Morrison-Woodbury formula to rational matrix functions.2011-11-05