Structure theory of reciprocal pairs of linear transformations (Q1334907)

If \(A\), \(B\) is a nonsingular reciprocical pair of linear transformations (i.e. acting between two vector spaces \(V\) and \(W\) in opposite directions), then the Jordan canonical forms of \(BA\) and \(AB\) are the same. This Jordan form \(J\) serves to classify completely the nonsingular pairs. Moreover, bases exist in the vector spaces \(V\) and \(W\) such that with respect to these bases both \(A\) and \(B\) appear as the matrix \(J^{1/2}\). Further one studies the cases of nilpotent reciprocical pairs. An illustrative application is also given.