A note on the eigenvalues of double band matrices
  We consider matrices containing two diagonal bands of positive entries. We show that all eigenvalues of such matrices are of the form $r \zeta$, where $r$ is a nonnegative real number and $\zeta$ is a $p$th root of unity, where $p$ is the period of the matrix, which is computed from the distance between the bands. 
Tyler McMillen