Can we relate this class of matrices to circulant matrices? I was having a look into circulant matrices, in particular very simple ones where, for each column, only one entry of the matrix is one. e.g. for a simple 3-dimensional case:
$$
A=\begin{bmatrix}
0 & 0 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0 \\
\end{bmatrix}
$$
Describing a dynamical system $$\dot{x}=Af(x)-x$$ where $f(x)$ is a properly defined elementwise operator on the vector x one can think of the dynamics above as the dynamics of interconnected species in a 'positive loop' plus a natural degradation. There are different examples in the literature analyzing this type of simple models. I know that $A$ is a circulant matrix and its properties (eigenvalues, eigenvectors, etc ...) are well known. 
But what happens when one considers a negative loop (making one or more rows negative)? To be more specific, I am interested in all these matrices that can be obtained from circulant matrices, by changing the sign of some of the rows.
For example consider the matrix:
$$
A^{'}=\begin{bmatrix}
0 & 0 & -1 \\
1 & 0 & 0 \\
0 & 1 & 0 \\
\end{bmatrix}
$$
how do we classify this kind of matrices? Or, can we analyse these non-circulant matrices using the properties of the circulant matrices? 
For example one can obtain $A^{'}$ by left-multiplying $A$ with an appropriate diagonal matrix but I can not go much further than this. Do you have any suggestion/reference?
 A: If an $n\times n$ circulant matrix has only one nonzero entry per column, then it is a scalar multiple of a permutation matrix corresponding to a cyclic permutation of length $n$:
$$ \begin{pmatrix} 
0 & 0 & 0 & c \\ c & 0 & 0 & 0 \\ 0 & c & 0 & 0 \\ 0 & 0 & c & 0 
\end{pmatrix} = 
c \begin{pmatrix}
0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 
\end{pmatrix}
$$
If we wanted to "classify" matrices which are similar in having the same sparse pattern of nonzero entries as the previously considered circulant matrix, but allow arbitrary nonzero values in those entries, one might express this as an invertible diagonal matrix $D$ times a corresponding circulant permutation matrix $P$:
$$ M = DP $$

Let's turn to the consideration of eigenvalues and eigenvectors.  For the original matrix, with no negative entries:
$$ A=\begin{pmatrix}
0 & 0 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0 \\
\end{pmatrix} $$
the eigenvalues are the roots of its characteristic polynomial:
$$ f_A(x) = x^3 - 1 $$
Note that these eigenvalues satisfy $x^3 = 1$, i.e. the cubic roots of unity.  One of them is real, $x=1$, and the other two roots are a pair of complex conjugates, $x = (-1 \pm i\sqrt 3)/2$.
On the other hand if we change an entry to $-1$, we get:
$$ B=\begin{pmatrix}
0 & 0 & -1 \\
1 & 0 & 0 \\
0 & 1 & 0 \\
\end{pmatrix} $$
whose characteristic polynomial is:
$$ f_B(x) = x^3 + 1 $$
and whose eigenvalues are the roots $x=-1$ and $x = (1 \pm i\sqrt 3)/2$.
If we consider all $2^3 = 8$ possible sign choices of the nonzero entries, it turns out that all have characteristic polynomial $f_A(x)$ or $f_B(x)$.  Indeed those matrices with an even number of negative entries (including none) have characteristic polynomial $f_A(x)$ while those with an odd number of negative entries have characteristic polynomial $f_B(x)$.
To see why, let's first notice the connection with Frobenius companion matrices.  Specifically matrix $A$ is the companion matrix for $f_A(x)$ and matrix $B$ is for $f_B(x)$.
Moreover all eight of these $3\times 3$ "signed" permutations are easily seen to be similar to one of $A,B$.  For example,
$$ DAD = \begin{pmatrix}
0 & 0 & 1 \\
-1 & 0 & 0 \\
0 & -1 & 0 \\
\end{pmatrix} \text{ where } D = \begin{pmatrix}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 1 \\
\end{pmatrix} $$
Note that this $D$ is a symmetric orthogonal matrix, so the above matrix is similar to $A$.  Replacing $D$ with other diagonal matrices which have $\pm 1$ entries (on the diagonal) yields the additional similarity relations we want.
Note as well that the transpose of a matrix shares its characteristic polynomial and eigenvalues.   Next we will take up the argument for larger versions of these circulant permutation related matrices.
