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

Polynomial n × n matrices A(x) and B(x) over a field mathbbF mathbb{F} are called semiscalar equivalent if there exist a nonsingular n × n matrix P over mathbbF mathbb{F} and an invertible n × n matrix Q(x) over mathbbF mathbb{F} [x] such that A(x) = PB(x)Q(x). We give a canonical form with respect to semiscalar equivalence for a matrix pencil A(x) = A _0x - A ₁, where A ₀ and A ₁ are n × n matrices over mathbbF mathbb{F} , and A ₀ is nonsingular.