the eigenvalues of a generalized circulant matrix A $2k\times 2k$ circulant matrix $\ C$ takes the form 
\begin{align}
    C= \begin{bmatrix} c_0 & c_{2k-1} & \dots & c_{2} & c_{1} \\ 
c_{1} & c_0 & c_{2k-1} & & c_{2} \\ 
\vdots & c_{1}& c_0 & \ddots & \vdots \\ 
c_{2k-2} & & \ddots & \ddots & c_{2k-1} \\ 
c_{2k-1} & c_{2k-2} & \dots & c_{1} & c_0 \\ \end{bmatrix}. 
\end{align}
Now we consider a $2k\times 2k$ generalized circulant matrix $\ C_{1}$
\begin{align}
    C_{1}= \begin{bmatrix} c_0 & -c_{2k-1} & \dots & c_{2} & -c_{1} \\ 
c_{1} & -c_0 & c_{2k-1} & & -c_{2} \\ 
\vdots & -c_{1}& c_0 & \ddots & \vdots \\ 
c_{2k-2} & & \ddots & \ddots & -c_{2k-1} \\ 
c_{2k-1} & -c_{2k-2} & \dots & c_{1} & -c_0 \\ \end{bmatrix}. 
\end{align}
That is the first column shifts with a multiplication of $-1$.
Then what are the eigenvalues of $\ C_{1}$? Thank you!
 A: $C_1$ isn't quite a "generalized $2k \times 2k$ circulant",
because it does not include $C$ as a special case.  It is, however,
block-circulant: let $A_0,\ldots,A_{k-1}$ be the $2\times 2$ matrices
$$
A_j = \left[\begin{array}{cc}
   c_{2j} & -c_{2j-1} \\ c_{2j+1} & -c_{2j}
\end{array}\right]
\quad(j=0,1,\ldots,k-1)
$$
(with $c_{-1}$ interpreted as $c_{2k-1}$); then $C_1$ is a $k\times k$
block-circulant with first column $A_0,\ldots,A_{k-1}$.  As with
ordinary circulant matrices, a block-circulant matrix is conjugate
over $\bf C$ to a block-diagonal matrix with $n$-th block
$$
\hat A_n := \sum_{j=0}^{k-1} e^{2\pi i nj/k} A_j.
$$
So the eigenvalues of $C_1$ are the eigenvalues of the $\hat A_n$,
which in this $2 \times 2$ case are given by the formula
$\frac12(t \pm \sqrt{t^2-4\Delta})$ where $t$ is the trace and
$\Delta$ is the determinant.
In the present case, each $A_j$ has trace zero, so the same is true of
all the $\hat A_n$, and the eigenvalues are simply
$\pm \sqrt{-\det \hat A_n}$.
This suggests an alternative description.  Form the matrix $C_1^2$,
and note that its entries are zero when the row and column index
have opposite parity.  Separating odd and even indices we find that
$C_1^2$ is conjugate with a block-diagonal matrix
$\left[{C \; 0 \atop 0 \; C}\right]$ for some $k \times k$ matrix $C$
that is circulant in the usual sense.  Compute the eigenvalues of this
circulant matrix $C$ as usual, and then extract their square roots
to recover the eigenvalues of $C_1$.
