Sum/multiplication of two circulant matrices is a circulant matrix
I know that the sum/multiplication of two circulant matrices is a circulant matrix. I'm looking for the shortest/easiest way to prove those two theorems. I could represent $A$ and $B$ as full $n\times n$ matrices and prove it easily enough by looking at $a_{ij}+b_{ij}$. But is there a shorter proof?
Solution 1:
A matrix is circulant if and only if it commutes with $$J=\pmatrix{0&1&0&\cdots&0\\ 0&0&1&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&1\\ 1&0&0&\cdots&0}$$ From this observation, it is apparent that sums and products of circulants are circulant.