# Triangularization of matrix polynomials

Taslaman, Leo and Tisseur, Francoise and Zaballa, Ion (2012) Triangularization of matrix polynomials. [MIMS Preprint]

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## Abstract

For an algebraically closed field $\F$, we show that any matrix polynomial $P(\l)=\sum_{j=0}^\ell \l^jA_j$ with $A_j\in\F^{\nbym}$, $n\le m$, can be reduced over $\F[\l]$ to an $\nbym$ upper triangular matrix polynomial of grade $\ell$ preserving the finite and infinite elementary divisors. We also characterize the real matrix polynomials that are triangularizable over the real numbers and show that those that are not triangularizable are quasi-triangularizable with diagonal blocks of sizes $1\times 1$ and $2\times 2$.

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 09 Aug 2012 08 Nov 2017 18:18 http://eprints.maths.manchester.ac.uk/id/eprint/1857

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