The eigenvectors and eigenvalues of block circulant matrices had been found
for real symmetric matrices with symmetric submatrices, and for block circulant
matrices with circulant submatrices. The eigenvectors are now found
for general block circulant matrices, including the Jordan Canonical Form
for defective eigenvectors. That analysis is applied to Stephen J. Watson’s
alternating circulant matrices, which reduce to block circulant matrices with
square submatrices of order 2.
Tee, G.J. (2005), Eigenvectors of block circulant and alternating circulant matrices, Research Letters in the Information and Mathematical Sciences, 8, 123-142