A classification of upper equivalent matrices. The generic case (Q1179077)

Two \(n\times n\) complex matrices \(A\) and \(B\) are called upper equivalent iff there is a nonsingular lower triangular matrix \(L\) such that \(B\) and \(L^{-1}AL\) have the same upper triangular parts. The authors obtain a set of invariants for this equivalence relation. The problem considered may be formulated as follows. Given a matrix \(A\) and the upper triangular part of a matrix \(B\) when do there exist a completion of \(B\) and a nonsingular lower triangular \(L\) such that \(B=L^{-1}AL\)? This question is answered under the assumption that any minor of a submatrix in the upper triangular part of \(A\) is nonzero.