Canonical form with respect to semiscalar equivalence for a matrix pencil with nonsingular first matrix

Abstract

Polynomial matrices $A(x)$ and $B(x)$ of size $n \times n$ over a field $\mathbb{F}$ are called semiscalar equivalent if there exist a nonsingular $n \times n$ matrix $P$ over $\mathbb{F}$ and an invertible $n \times n$ matrix $Q(x)$ over $\mathbb{F}[x]$ such that $A(x) = PB(x)Q(x)$. We give a canonical form with respect to the semiscalar equivalence for a matrix pencil $A(x) = A_0x - A_1$, where $A_0$ and $A_1$ are $n \times n$ matrices over $\mathbb{F}$, and $A_0$ is nonsingular.