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.