# Quasi-triangularization of matrix polynomials over arbitrary fields

Anguas, Luis M. and Dopico, Froilán M. and Hollister, Richard and Mackey, D. Steven (2021) Quasi-triangularization of matrix polynomials over arbitrary fields. [MIMS Preprint]

In \cite{TasTisZab}, Taslaman, Tisseur, and Zaballa show that any regular matrix polynomial $P(\la)$ over an algebraically closed field is spectrally equivalent to a triangular matrix polynomial of the same degree. When $P(\la)$ is real and regular, they also show that there is a real quasi-triangular matrix polynomial of the same degree that is spectrally equivalent to $P(\la)$, in which the diagonal blocks are of size at most $2 \times 2$. This paper generalizes these results to regular matrix polynomials $P(\la)$ over arbitrary fields $\bF$, showing that any such $P(\la)$ can be quasi-triangularized to a spectrally equivalent matrix polynomial over $\bF$ of the same degree, in which the largest diagonal block size is bounded by the highest degree appearing among all of the $\bF$-irreducible factors in the Smith form for $P(\la)$.