On the equivalence of polynomial matrices over a field - Prokip , V. (Ukrains’kyi Matematychnyi Zhurnal, 2024)
UDC 512.64The polynomial $(n\times n)$ matrices $A(\lambda )$ and $B(\lambda)$ over a field ${\mathbb F}$ are  called semiscalar equivalent if there exists a nonsingular $(n\times n)$ matrix $P$ over ${\mathbb F}$ and  an invertible $(n\times n)$  polynomial matrix $Q(\lambda )$ over ${\mathbb F[\lambda}]$ such that $A(\lambda ) = PB(\lambda )Q(\lambda )$.  We establish conditions under which nonsingular polynomial matrices $A(\lambda )$ and $B(\lambda )$  are semiscalar equivalent.  As a consequence, we present the conditions of equivalence and similarity of two sets of $(n\times n)$  matrices over an arbitrary field ${\mathbb F}.$