We can solve this equation in an iterative manner. We let $P_k(\lambda)$ be the value of $k$ subdeterminant of the above matrix of dimension $n\times n$. The polynomial $P_k(\lambda)$ is clearly a polynomial of degree $k$. Starting with $P_1(\lambda)$ we have $P_1(\lambda)=d_1-\lambda$. The next polynomial reads $P_2(\lambda)=(d_2-\lambda)P_1(\lambda)-e_1^2$. By expanding the determinant for $P_k(\lambda)$ in terms of the minors of the $n$th column we arrive at the recursion relation $$P_k(\lambda)=(d_k-\lambda)P_{k-1}(\lambda)-e_{k-1}^2P_{k-2}(\lambda).$$ Together with the starting values $P_1(\lambda)$ and $P_2(\lambda)$ and good root searching methods we arrive at an efficient computational scheme for finding the roots of $P_n(\lambda)$. However, for large matrices this algorithm is rather inefficient and time-consuming.