  We study some aspects of the invariant pair problem for matrix polynomials,
as introduced by Betcke and Kressner and by Beyn and Thuemmler. Invariant pairs
extend the notion of eigenvalue-eigenvector pairs, providing a counterpart of
invariant subspaces for the nonlinear case. We compute formulations for the
condition numbers and backward errors of invariant pairs and solvents. We then
adapt the Sakurai-Sugiura moment method to the computation of invariant pairs,
including some classes of problems that have multiple eigenvalues. Numerical
refinement via a variants of Newton's method is also studied. Furthermore, we
investigate the relation between the matrix solvent problem and the
triangularization of matrix polynomials.
