Several properties of invariant pairs of nonlinear algebraic eigenvalue problems

We analyze several important properties of invariant pairs of nonlinear algebraic eigenvalue problems of the form T (λ)v = 0. Invariant pairs are generalizations of invariant subspaces in association with block Rayleigh quotients of square matrices to a nonlinear matrix-valued function T (·). They play an important role in the analysis of nonlinear eigenvalue problems and algorithms. In this paper, we first show that the algebraic, partial, and geometric multiplicities together with the Jordan chains corresponding to an eigenvalue of T (λ)v = 0 are completely represented by the Jordan canonical form of a simple invariant pair that captures this eigenvalue. We then investigate approximation errors and perturbations of a simple invariant pair. We also show that second order accuracy in eigenvalue approximation can be achieved by the two-sided block Rayleigh functional for non-defective eigenvalues. Finally, we study the matrix representation of the Frechet derivative of the eigenproblem, and we discuss the norm estimate of the inverse derivative, which measures the conditioning and sensitivity of simple invariant pairs.