I am new to optimization and I would like to know if there are any results/methods for the following optimization problem:
$$\text{Minimize: }T = \sum_{k=2}^N \frac{1}{1-\lambda_k}\text{ subject to: } \sum_{k=2}^N \lambda_k = -1 \text{ and } |\lambda_k| \leq 1$$
It worth mention that $\lambda_k$ are the eigenvalues of a symmetric matrix.
Can I say something about the distribution of $\lambda_k$ that minimize my objective function?
Does this problem follow any special class of problems? Where should I look for more information on this type of problems?