The eigenvalues would really only reveal how many classes (single classifier) or how many classifiers are correlated with one another (multiple classifiers).  But if you look at the Kappa matrix for all possible pairs of classifiers applied to the 3-class Wine dataset below:

[![enter image description here][1]][1]

You can then quasi-diagonalize the Kappa matrix and perform hierarchical cluster analysis using a modified form of Euclidean distance to see which classifiers cluster with one another:

[![enter image description here][2]][2]  

Note, this Kappa matrix is a $13 \times 13$, so when looking at the clustering, I notice three large groups of classifiers which are sizes 6, 5, and 2, so I would expect the eigenvalues of a correlation matrix (MCC) to perhaps show something similar to this. That is, you wouldn't likely have one large eigenvalue like 8 or 9, with the remaining in the other dimensions.  

I believe you can do more surgery (drilling down) regarding discovery on classifier behavior taking this approach instead of perform eigendecomposition.  


  [1]: https://i.sstatic.net/r63rA.png
  [2]: https://i.sstatic.net/FTPsz.png