# Triangularization of matrix polynomials

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

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 $2\times 2$.