I have the following matrix:
$$\mathbf{M} = \begin{pmatrix} G_{1}^{(N)} & G_{2}^{(N)} \\ G_{2}^{(N)} & G_{3}^{(N)} \end{pmatrix}$$,
where $G^{(N)}_{j}$ are symmetric circulant matrices of size $N-\text{by}-N$ with complex-valued entries. Can this be diagonalized (not block-diagonalized!)?
This is not a full diagonalization, but we can end up with a block-diagonalization where each block has size only $2 \times 2$.
Let $F$ denote the DFT matrix of size $N$. Denote $$ P_1 = I_2 \otimes F = \pmatrix{F & 0\\0 & F}. $$ Here, $\otimes$ denotes the Kronecker product. Note that $P$ is unitary, which is to say that $P^{-1} = P^*$ (the conjugate-transpose of $P$). We find that $$ P_1^{-1}MP_1 = \pmatrix{F^{-1}G_1^{(N)}F & F^{-1}G_2^{(N)}F\\ F^{-1}G_2^{(N)}F & F^{-1}G_3^{(N)}F} = \pmatrix{D_1 & D_2\\D_2 & D_3}, $$ and each of the matrices $D_1,D_2,D_3$ is diagonal. Now, let $P_2$ denote the commutation matrix $P_2 = K^{(2,N)}$. $P_2$ is a symmetric permutation matrix, so that $P^{-1} = P^T$. We have $$ P_2^{-1}(P_1^{-1}MP_1)P_2 = \pmatrix{A_1\\ & \ddots \\ && A_N}. $$ Each matrix $A_j$ has size $2 \times 2$. If $D_{jk}$ denotes the $k$th diagonal entry of $D_j$, then we have $$ A_k = \pmatrix{D_{1k} & D_{2k}\\ D_{2k} & D_{3k}}. $$ $M$ is diagonalizable if and only if this last matrix $P_2^{-1}P_1^{-1}MP_1P_2$ is diagonalizable, which holds if and only if each matrix $A_k$ is diagonalizable. We can see that the matrices $A_k$ are symmetric with complex entries, but this does not guarantee that they are diagonalizable.