  We present a Bayesian scheme for the approximate diagonalisation of several
square matrices which are not necessarily symmetric. A Gibbs sampler is derived
to simulate samples of the common eigenvectors and the eigenvalues for these
matrices. Several synthetic examples are used to illustrate the performance of
the proposed Gibbs sampler and we then provide comparisons to several other
joint diagonalization algorithms, which shows that the Gibbs sampler achieves
the state-of-the-art performance on the examples considered. As a byproduct,
the output of the Gibbs sampler could be used to estimate the log marginal
likelihood, however we employ the approximation based on the Bayesian
information criterion (BIC) which in the synthetic examples considered
correctly located the number of common eigenvectors. We then succesfully
applied the sampler to the source separation problem as well as the common
principal component analysis and the common spatial pattern analysis problems.
