# Deflating Quadratic Matrix Polynomials with Structure Preserving Transformations

Tisseur, Francoise and Garvey, Seamus D. and Munro, Christopher (2009) Deflating Quadratic Matrix Polynomials with Structure Preserving Transformations. [MIMS Preprint]

Given a pair of distinct \e s $(\l_1,\l_2)$ of an $\nbyn$ quadratic matrix polynomial $Q(\l)$ with nonsingular leading coefficient and their corresponding \ev s, we show how to transform $Q(\l)$ into a quadratic of the form $\twobytwoa{\Qd(\l)}{0}{0}{q(\l)}$ having the same \e s as $Q(\l)$, with $\Qd(\l)$ an $(n-1)\times (n-1)$ quadratic matrix \py\ and $q(\l)$ a scalar quadratic \py\ with roots $\l_1$ and $\l_2$. This block diagonalization cannot be achieved by a similarity transformation applied directly to $Q(\l)$ unless the \ev s corresponding to $\l_1$ and $\l_2$ are parallel. We identify conditions under which we can construct a family of $2n\times 2n$ elementary similarity transformations that (a) are rank-two modifications of the identity matrix, (b) act on linearizations of $Q(\l)$, (c) preserve the block structure of a large class of block symmetric linearizations of $Q(\l)$, thereby defining new quadratic matrix polynomials $Q_1(\l)$ that have the same \e s as $Q(\l)$, (d) yield quadratics $Q_1(\l)$ with the property that their \ev s associated with $\l_1$ and $\l_2$ are parallel and hence can subsequently be deflated by a similarity applied directly to $Q_1(\l)$. This is the first attempt at building elementary transformations that preserve the block structure of widely used linearizations and which have a specific action.