Polynomial supersymmetry for matrix Hamiltonians

► Constructing of any matrix intertwining operator in terms of transformation functions. ► Polynomial SUSY exists for any matrix intertwining operator of minimal order. ► Minimization of a matrix intertwining operator and criterion of minimizability. ► (Ir)reducibility of a matrix intertwining operator and criterion of reducibility. ► Absolutely irreducible matrix intertwining operators exist in contrast to scalar case.