I'm looking to find a closed-form expression for the norm (max singular value) of a "sign-flipped circulant matrix", in particular, an n by n matrix of the form
i.e., each row is the preceding row shifted to the right, with the entry flipping in sign. Precisely, C is given by $C_n = (t_{y-x})_{x,y=1}^n$ with \begin{equation} t_l = \begin{cases} \phantom{-} 1, & \text{if } \vert l \vert \leq k -1 \; \; \vee \; \; l = k,\\ -1, & \text{if } \vert l \vert \geq n-k+1 \; \; \vee \; \; l = -k, \\ 0 & \text{else} \end{cases} \end{equation}
C^T C is also such a matrix (although not with {0,1,-1} entries), so any reference to the eigenvalues of such matrices would be very helpful.