Suppose $Q$ is a dot product of diagonal matrix A and matrix B: $$ Q=A\cdot B= \left( \begin{matrix} a_1 & 0 & \cdots & 0 \\ 0 & a_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a_n \end{matrix} \right) \cdot \left( \begin{matrix} b_{11} & b_{12} & \cdots & b_{1n} \\ b_{21} & b_{22} & \cdots & b_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ b_{n1} & b_{n2} & \cdots & b_{nn} \end{matrix} \right), $$ $a_i>0$, $b_{ij}>0$ if $i\neq j$, $b_{ii}=-\sum_{j\neq i}{b_{ij}}$.
Let's assume that B is diagonalizable, so $B=P D P^{-1}$, where $D$ is diagonal matrix.
Can we easily compute $e^Q$ if we have computed already $P$, $D$ and $e^B$? Or we have to perform Jordan decomposition for every new matrix $A$?
P.S. AFAIU it is possible to find $e^{Qt}$ where $t$ is scalar: $e^{Qt}=P e^{Dt} P^{-1}$. So I was wondering whether the same is possible with diagonal matrix.
