Approximation of Eigenvalues

Abstract

The computation of the eigenvalues of a square matrix is a problem of considerable difficulty. The naive idea, according to which it is enough to compute the characteristic polynomial and then find its roots, turns out to be hopeless because of Abel’s theorem, which states that the general equation P(x) = 0, where P is a polynomial of degree d ≥ 5, is not solvable using algebraic operations and roots of any order. For this reason, there exists no direct method, even an expensive one, for the computation of Sp(M).