Quasi-Triangularization of Matrix Polynomials over Arbitrary Fields

In [19], Taslaman, Tisseur, and Zaballa show that any regular matrix polynomial $P(\lambda)$ over an algebraically closed field is spectrally equivalent to a triangular matrix polynomial of the same degree. When $P(\lambda)$ 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 $P(\lambda)$, in which the diagonal blocks are of size at most $2 \times 2$. This paper generalizes these results to regular matrix polynomials $P(\lambda)$ over arbitrary fields $\mathbb{F}$, showing that any such $P(\lambda)$ can be quasi-triangularized to a spectrally equivalent matrix polynomial over $\mathbb{F}$ of the same degree, in which the largest diagonal block size is bounded by the highest degree appearing among all of the $\mathbb{F}$-irreducible factors in the Smith form for $P(\lambda)$.