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Assume I am given an $n \times n$ matrix $A$ with real or complex coefficients. Its matrix exponential is denoted by $\exp(A)$ and is calculated as usual. Assume further that I want to rescale the first row $(a_{11}, \ldots, a_{1n})$ by a factor $r_1$, so it reads $(r_1 \cdot a_{11}, \ldots, r_1 \cdot a_{1n})$. When I do this for all other rows, I obtain a new matrix $B = (r_i \cdot a_{ij})_{i,j=1}^n$.

Knowing $A,B$ and $\exp(A)$, is there a nice way to express $\exp(B)$?

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    $\begingroup$ Let $\Delta$ be the diagonal matrix with entries $r_i$, so that $B = \Delta A$. Then $\exp B = \exp (\Delta A)$. So the answer is almost certainly no, there is no nice way to do it---even in the unlikely case that $\Delta A = A \Delta$. $\endgroup$ – David Handelman Jun 25 '17 at 14:28

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