I am attempting to implement a pairs trading algorithm for two securities by approximating a discretized version of the Ornstein-Uhlenbeck process:
\begin{equation*} 
d\mathbf{S}_t = \mathbf{\kappa}(\mathbf{\theta} - \mathbf{S}_t)dt + \mathbf{\sigma}d\mathbf{W}_t,
\end{equation*}
 using the vector autoregression:
\begin{equation*} 
\mathbf{S}_t = \mathbf{A} + \mathbf{BS}_{t-1} + \mathbf{\epsilon}_t.
\end{equation*}

Recall that the ideal co-integration factor corresponds to the eigenvector of $\kappa = \mathbf{I-B}$ with the largest corresponding eigenvalue.  However, I am encountering an empirical situation in which $\kappa$ is not diagonalizable over $\mathbb{R}$ and has only *complex* eigenvalues and eigenvectors.

This situation does not seem to be treated in any of the literature, even though there appears to be no reason, *a priori*, that it should not arise.  What are best practices for handling this situation, i.e., computing the co-integration factor when $\kappa$ is not diagonalizable?