  The paper presents methods of eigenvalue localisation of regular matrix
polynomials, in particular, stability of matrix polynomials is investigated.
For this aim a stronger notion of hyperstability is introduced and widely
discussed. Matrix versions of the Gauss-Lucas theorem and Sz\'asz inequality
are shown. Further, tools for investigating (hyper)stability by multivariate
complex analysis methods are provided. Several second- and third-order matrix
polynomials with particular semi-definiteness assumptions on coefficients are
shown to be stable.
