Suppose the matrices $A_N$ which in dimension $N$ have non-zero elements given by
$(A_N)_{i,i+1}=i$ and $(A_N)_{i+1,i}=N-i$, for $i=1,...,N-1$
For example,
$A_4=\left[ \begin {array}{cccc} 0&1&0&0\\ 3&0&2&0 \\0&2&0&3\\ 0&0&1&0\end {array} \right] $, $A_5=\left[ \begin {array}{ccccc} 0&1&0&0&0\\ 4&0&2&0&0\\ 0&3&0&3&0\\ 0&0&2&0&4\\ 0&0&0&1&0\end {array} \right] $
Matrix $A_N$ is equal to $N-1$ times a stochastic matrix, so the Perron-Frobenius theorem guarantees that $N-1$ is the largest eigenvalue, with left eigenvector constant.
However, these matrices are very particular. For example, the eigenvalues are precisely the integers from $-(N-1)$ to $(N-1)$. Moreover, the right eigenvectors $v_i$ are such that the generating functions $f_i(x)=\sum_{k=1}^N(v_i)_k x^k$ are given by $f_i(x)=x(1+x)^{N-i}(x-1)^{i-1}$.
I could not get a handle on the left eigenvectors.
My questions are: Has anyone seen these matrices before? Does anyone have a suggestion on how to understand the left eigenvectors?