# Triangularizing Quadratic Matrix Polynomials

Tisseur, Françoise and Zaballa, Ion (2012) Triangularizing Quadratic Matrix Polynomials. [MIMS Preprint]

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## Abstract

We show that any regular quadratic matrix polynomial can be reduced to an upper triangular quadratic matrix polynomial over the complex numbers preserving the finite and infinite elementary divisors. We characterize the real quadratic matrix polynomials that are triangularizable over the real numbers and show that those that are not triangularizable over the real numbers are quasi-triangularizable with diagonal blocks of sizes $1\times 1$ and $2 \times 2$. We also derive complex and real Schur-like theorems for linearizations of quadratic matrix polynomials with nonsingular leading coefficients. In particular, we show that for any monic linearization $\l I+A$ of an $n\times n$ quadratic matrix polynomial, there exists a nonsingular matrix defined in terms of $n$ orthonormal vectors that transforms $A$ to a companion linearization of a (quasi)-triangular quadratic matrix polynomial. This provides the foundation for designing numerical algorithms for the reduction of quadratic matrix polynomials to upper (quasi)-triangular form.

Item Type: MIMS Preprint MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theoryMSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis Dr Françoise Tisseur 20 Aug 2012 08 Nov 2017 18:18 http://eprints.maths.manchester.ac.uk/id/eprint/1861

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