The eigenvectors and eigenvalues of a Circulant matrix are well-known to be related to the discrete Fourier transform of entries of one row (the exact terms are given here).
My question: is there any result regarding the eigenvectors and eigenvalues of an incomplete Circulant matrix, namely, a matrix which contains the first $m<n$ rows of some $n\times n$ Circulant matrix?
Thank you!
A companion matrix (see any linear algebra book) is an example for the case $m=n-1$.