For $n=3,$ $L$ matrix is as below:
$$ \left[\begin{matrix} 1&0&0\\ -1&1&0\\ -0.5&-0.5&1\\ \end{matrix}\right] $$
for $n=4$ we have:
$$ \left[\begin{matrix} 1&0&0&0\\ -1&1&0&0\\ -0.5&-0.5&1&0\\ -1/3&-1/3&-1/3&1 \end{matrix} \right] $$
and for $N$th we have: $$ \left[ \begin{matrix} 1&0&0&0&\cdots&0&0\\ -1&1&0&0&\cdots&0&0\\ -0.5&-0.5&1&0&\cdots&0&0\\ \vdots&\vdots&\vdots&\vdots& & \vdots&\vdots\\ -1/(N-2)&-1/(N-2)&-1/(N-2)&-1/(N-2)&\cdots&1&0\\ -1/(N-1)&-1/(N-1)&-1/(N-1)&-1/(N-1)&\cdots&-1/(N-1)&1 \end{matrix} \right] $$
How to get the roots (eigenvalues) of the $N$th order of the $L+L^T$ matrix? is there a specific logic between the roots of this matrix?