ABSTRACT
For what follows the reader will need the knowledge of de…nitions and properties of the greatest common (left, right) divisors of the matrix polynomials, of the unimodular matrix polynomials, as well as the knowledge of the conditions for their (left, right) coprimeness (see the books by J. P. Antsaklis and A. N. Michel [8, pp. 526-528, 535-540], C.-T. Chen [29, pp. 591-599] and T. Kailath [198, pp. 373-382]). A rational matrix function M(:) = M1D (:)MN (:) [M(:) = MN (:)M
1 D (:)] is
irreducible if, and only if, its polynomial matrices MD(:) and MN (:) are (left and/or right) coprime (see C.-T. Chen [29, pp. 591-599] and T. Kailath [198, pp. 373-382]). The greatest common (left L(:), also right R(:)) divisor ofMD(:) and of MN (:) cancels itself in M(:), even though L(:) and R(:) are unimodular polynomial matrices,
MD(s) = L(s)D(s); MN (s) = L(s)N(s) =) M(:) =M1D (:)MN (:) = D
1(s)L1(s)L(s)N(s) = D1(s)N(s);
MD(s) = D(s)R(s); MN (s) = N(s)R(s) =) M(:) =MN (:)M
1 D (:) = N(s)R(s)R
1(s)D1(s) = N(s)D1(s):
In [81], [148] it was noted that such a de…nition of the irreducibility is not fully adequate for MIMO systems. It was the reason to show that an irreducible complex matrix function can be nondegenerate or degenerate in the following sense [81], [148]:
De…nition 93 A rational matrix function M(:) =M1D (:)MN (:) [respectively, M(:) = MN (:)M
1 D (:)] is
a) row nondegenerate if, and only if, respectively: (i) the greatest common left [right] divisor of MD(:) and of MN (:) is a
nonsingular constant matrix, and
(ii) the greatest common scalar factors of detMD(s) and of all elements of every row of (adjMD(s))MN (s) [respectively, of all elements of every row of MN (s) (adjMD(s))] are nonzero constants that can be mutually di¤erent.
