  We study intertwining relations for matrix one-dimensional, in general,
non-Hermitian Hamiltonians by matrix differential operators of arbitrary order.
It is established that for any matrix intertwining operator Q_N^- of minimal
order N there is a matrix operator Q_{N'}^+ of different, in general, order N'
that intertwines the same Hamiltonians as Q_N^- in the opposite direction and
such that the products Q_{N'}^+Q_N^- and Q_N^-Q_{N'}^+ are identical
polynomials of the corresponding Hamiltonians. The related polynomial algebra
of supersymmetry is constructed. The problems of minimization and of
reducibility of a matrix intertwining operator are considered and the criteria
of minimizability and of reducibility are presented. It is shown that there are
absolutely irreducible matrix intertwining operators, in contrast to the scalar
case.
