Deflating quadratic matrix polynomials with structure preserving transformations

Abstract

Given a pair of distinct eigenvalues ( λ 1 , λ 2 ) 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 Q d ( λ ) 0 0 q ( λ ) having the same eigenvalue s as Q ( λ ) , with Q d ( λ ) an ( n - 1 ) × ( n - 1 ) quadratic matrix polynomial and q ( λ ) a scalar quadratic polynomial with roots λ 1 and λ 2 . This block diagonalization cannot be achieved by a similarity transformation applied directly to Q ( λ ) unless the eigenvectors corresponding to λ 1 and λ 2 are parallel. We identify conditions under which we can construct a family of 2 n × 2 n 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 Q 1 ( λ ) that have the same eigenvalue s as Q ( λ ) , (d) yield quadratics Q 1 ( λ ) with the property that their eigenvectors associated with λ 1 and λ 2 are parallel and hence can subsequently be deflated by a similarity applied directly to Q 1 ( 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.