I have a sort of reverse problem, and I'm not sure if there is a simple solution.
I have a tridiagonal Hermitian matrix: $$ A = \begin{bmatrix} 0 & a_1 & 0 & 0 & 0 \\ a_1 & 0 & a_2 & 0 & 0 \\ 0 & a_2 & 0 & a_3 & 0 \\ 0 & 0 & a_3 & 0 & a_n \\ 0 & 0 & 0 & a_n & 0 \\ \end{bmatrix} $$ where $a_n$ are unknown; however, I know $A$'s eigenvalues: $$ \lambda = \{\lambda_1, \lambda_2, \lambda_3, \lambda_n\}. $$
I want to find the $a_n$'s that would give me the known eigenvalues that I have. So I'm after the matrix that gives me these eigenvalues. I also know the order of the eigenvalues, so I that, say, the first eigenvalue belongs to the largest eigenvector, and the second to the second largest eigenvector. I don't know the eigenvectors though, and I don't really care what they are.
I've looked into closed-form equations to calculate the characteristic polynomials for a matrix of the form of $A$; however, I'm not sure how easy it is to go from these to $a_n$'s.
Any advice would be much appreciated.