I have a special $N\times N$ matrix with the following form. It is symmetric and zero row (and column) sums. $$K=\begin{bmatrix} k_{11} & -1 & \frac{-1}{2} & \frac{-1}{3} & \frac{-1}{4} & \ldots & \frac{-1}{N-2} & \frac{-1}{N-1} & \\ -1 & k_{22} & \frac{-1}{2} & \frac{-1}{3} & \frac{-1}{4} & \ldots & \frac{-1}{N-2} & \frac{-1}{N-1} & \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \\ \frac{-1}{N-2} & \frac{-1}{N-2} & \frac{-1}{N-2} & \frac{-1}{N-2} & \frac{-1}{N-2} & \ldots & k_{N-1,N-1} & \frac{-1}{N-1} & \\ \frac{-1}{N-1} & \frac{-1}{N-1} & \frac{-1}{N-1} & \frac{-1}{N-1} & \frac{-1}{N-1} & \ldots & \frac{-1}{N-1} & 1 & \\ \end{bmatrix} $$ where $K_{ii}=\sum_{j=1, j\ne i}^{N}{(-k_{ij})}$ for $i=1, 2,3,\ldots , N $

For example if N=4, we have: $$K = \begin{bmatrix} 11/6 & -1 & -1/2 & -1/3 & \\ -1 & 11/6 & -1/2 & -1/3 & \\ -1/2 & -1/2 & 4/3 & -1/3 & \\ -1/3 & -1/3 & -1/3 & 1 & \\ \end{bmatrix} $$

How can I find an explicit equation for its eigenvalues?