Kloeden Platen Schurz 1994 states the linear 2D SDE:
$$dX_t = A X_t dt + B X_t dW_t$$
where $a=5$ and $b=0.01$ and
$$A = \Big( \begin{matrix} -a & a \\ a & -a \end{matrix} \Big)$$ $$B = \Big( \begin{matrix} b & 0 \\ 0 & b \end{matrix} \Big)$$
They give explicit solution
$$X(t) = X_0 \exp{\Big( \big(A - \frac{B^2}{2} \big) t + B W_t \Big)}$$
As an example starting point, they give $X_0=\big(\begin{matrix} 1 \\ 0\end{matrix}\big)$. However, at $t=0$, $W_t=0$ and
$$\Big( \big(A - \frac{B^2}{2} \big) 0 + B 0 \Big) = \Big( \begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix} \Big)$$
and $$\exp{\Big( \begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix} \Big)} = \Big( \begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix} \Big)$$
So
$$X(0) = X_0 \Big( \begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix} \Big) = \big(\begin{matrix} 1 \\ 0\end{matrix}\big) \Big( \begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix} \Big) = \big(\begin{matrix} 1 \\ 1\end{matrix}\big) \neq X_0 $$
Please help. I should be able to choose any starting point for $X_0$ and then evolve it according to the SDE. The given explicit solution seems wrong in this case. Is it wrong or am I missing something obvious?
It is pointed out in comments to this question that NumPy's exp and log are doing element-wise operations, which are incorrect in this case. I should have been using SciPy's expm and logm.