It is known that an \(n\times n\) structured matrix \(M\) can be associated with an appropriate displacement operator \(L\) such that the rank \(r\) of \(L(M)\) satisfies \(r\ll n\) and the \(n^2\) entries of \(L(M)\) can be represented via only \(2rn\) parameters. The authors present a general method to express \(M\) via \(L(M)\) under very mild nonsingularity assumptions. The method unifies the derivation of known formulae and gives new formulae, in particular for the tangential Nevalinna-Pick problems. It accelerates known solution algorithms. The authors obtain general new matrix representations in the confluent case and substantially improve the known estimates for the norm \(\| L^{-1}\| \) which is critical in computations based on the displacement approach.