ON THE SMITH NORMAL-FORM OF STRUCTURED POLYNOMIAL-MATRICES

The Smith normal form of a polynomial matrix D(s) = Q(s) + T(s) is investigated, where D(s) is "structured" in the sense that (i) the coefficients of the entries of Q(s) belong to a field K, (ii) the nonzero coefficients of the entries of T(s) are algebraically independent over K, and (iii) every minor of Q(s) is a monomial in s. Such matrices have been useful in the structural approach in control theory. It is shown that all the invariant polynomials except for the last are monomials in s and the last invariant polynomial is expressed in terms of the combinatorial canonical form (CCF) of a layered mixed matrix associated with D(s). On the basis of this, the Smith form of D(s) can be computed by means of an efficient (polynomial-time) matroid-theoretic algorithm that involves arithmetic operations in the base field K only.