On semiscalar and quasidiagonal equivalence of matrices (Q2730514)

Polynomial matrices \(A(x)\) and \(B(x)\) are called semiscalarly equivalent if \(SA(x)R(x)=B(x)\) for some invertible matrices \(S\) and \(R(x)\). The author considers the class of matrices \(A(x)\) with only two different invariant factors \(\varphi_1(x),\varphi_2(x)\) such that all the roots of the polynomial \(\varphi_2(x)/\varphi_1(x)\) are simple. Within this class, a criterion of semiscalar equivalence is found. It is also shown that the problem of semiscalar equivalence is wild. The proof is based on the reduction to the equivalence of numerical matrices implemented by block-diagonal factors.