Symbolic estimation of distances between eigenvalues of Hermitian operator and unitary orbits classification

We use symbolic expressions for traces of positive integer powers of a Hermitian operator (or, equivalently, coefficients of corresponding characteristic polynomial) to find solutions for the problems as follows: Factorization of characteristic polynomial that collects all eigenvalues that have the same multiplicity into one polynomial factor with coefficients rationally expressed through coefficients of original characteristic polynomial. Finding with any accuracy a minimal distance between eigenvalues, as well as a minimal and a maximal eigenvalues of the Hermitian operator, applying only rational functions of corresponding coefficients of the characteristic polynomial. Estimation of related rate of convergence. Classification of unitary orbits of Hermitian operators.