Deflating quadratic matrix polynomials with structure preserving transformations

Given a pair of distinct eigenvalues (λ1,λ2) of an n×n quadratic matrix polynomial Q(λ) with nonsingular leading coefficient and their corresponding eigenvectors, we show how to transform Q(λ) into a quadratic of the form having the same eigenvalue s as Q(λ), with Qd(λ) an (n-1)×(n-1) quadratic matrix polynomial and q(λ) a scalar quadratic polynomial with roots λ1 and λ2. This block diagonalization cannot be achieved by a similarity transformation applied directly to Q(λ) unless the eigenvectors corresponding to λ1 and λ2 are parallel. We identify conditions under which we can construct a family of 2n×2n elementary similarity transformations that (a) are rank-two modifications of the identity matrix, (b) act on linearizations of Q(λ), (c) preserve the block structure of a large class of block symmetric linearizations of Q(λ), thereby defining new quadratic matrix polynomials Q1(λ) that have the same eigenvalue s as Q(λ), (d) yield quadratics Q1(λ) with the property that their eigenvectors associated with λ1 and λ2 are parallel and hence can subsequently be deflated by a similarity applied directly to Q1(λ). This is the first attempt at building elementary transformations that preserve the block structure of widely used linearizations and which have a specific action.