Bounds for eigenvalues of matrix polynomials Original Research Article

Upper and lower bounds are derived for the absolute values of the eigenvalues of a matrix polynomial (or λ-matrix). The bounds are based on norms of the coefficient matrices and involve the inverses of the leading and trailing coefficient matrices. They generalize various existing bounds for scalar polynomials and single matrices. A variety of tools are used in the derivations, including block companion matrices, Gershgorinʹs theorem, the numerical radius, and associated scalar polynomials. Numerical experiments show that the bounds can be surprisingly sharp on practical problems.