Eigenvectors of block diagonal matrix with circulant blocks

Question

Let $A_i$ be an $n_i \times n_i$ circulant matrix with full rank.

Its eigenvalues and eigenvectors are well known ( $a_0+ a_1 \omega_i + \cdots+ a_{n_i-1} \omega_i^{n_i-1}$ and the columns of the $n_i \times n_i$ DFT matrix respectively, where $\omega_i$ is an $n_i^{th}$ root of unity and $A_i=circulant(a_0,\ldots, a_{n_i})$.)

Let $$A=blockdiag(A_1,\ldots,A_v)$$ be $n\times n$ with $n=n_1+\cdots+n_v.$

Then the eigenvalues of $A$ will be the union of the eigenvalues of the individual matrices. What about the eigenvectors? Especially when eigenvalues of $A_i$ and $A_j$ overlap?

Answer 1

Block diagonal matrices with circulant blocks are diagonalized by block diagonal matrices with DFT blocks.

So the eigenvectors are the eigenvectors of the blocks, filled out with zeros.