Deflating quadratic matrix polynomials with structure preserving transformations

作者：Tisseur Francoise^*; Garvey Seamus D; Munro Christopher

来源：Linear Algebra and Its Applications, 2011, 435(3): 464-479.

DOI：10.1016/j.laa.2010.06.028

• Quadratic eigenvalue problem
• Linearization
• Structure preserving transformation
• Deflation

Given a pair of distinct eigenvalues (lambda(1), lambda(2)) of an n x n quadratic matrix polynomial Q(lambda) with nonsingular leading coefficient and their corresponding eigenvectors, we show how to transform Q(lambda) into a quadratic of the form [Q(d)(lambda) 0 0 q(lambda)] having the same eigenvalues as Q(lambda), with Q(d)(lambda) an (n - 1) x (n - 1) quadratic matrix polynomial and q(lambda) a scalar quadratic polynomial with roots lambda(1) and lambda(2). This block diagonalization cannot be achieved by a similarity transformation applied directly to Q(lambda) unless the eigenvectors corresponding to lambda(1) and lambda(2) are parallel. We identify conditions under which we can construct a family of 2n x 2n elementary similarity transformations that (a) are rank-two modifications of the identity matrix, (b) act on linearizations of Q(lambda), (c) preserve the block structure of a large class of block symmetric linearizations of Q(lambda), thereby defining new quadratic matrix polynomials Q(1) (lambda) that have the same eigenvalues as Q(lambda), (d) yield quadratics Q(1) (lambda) with the property that their eigenvectors associated with lambda(1) and lambda(2) are parallel and hence can subsequently be deflated by a similarity applied directly to Q(1) (lambda). This is the first attempt at building elementary transformations that preserve the block structure of widely used linearizations and which have a specific action.

• 出版日期2011-8-1