On the Equivalence of Polynomial Matrices Over a Field

The polynomial (n × n) matrices A(λ) and B(λ) over a field 𝔽 are called semiscalar equivalent if there exist a nonsingular (n × n) matrix P over 𝔽 and an invertible (n × n) polynomial matrix Q(λ) over 𝔽[λ] such that A(λ) = PB(λ)Q(λ). We establish conditions under which nonsingular polynomial matrices A(λ) and B(λ) are semiscalar equivalent. As a consequence, we present the conditions of equivalence and similarity for two sets of (n × n) matrices over an arbitrary field 𝔽.