On the Smith Normal Form of Structured Polynomial Matrices, II
Kazuo Murota · SIAM Journal on Matrix Analysis and Applications · 1993
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 ${\bf K}$ and (ii) the nonzero coefficients of the entries of $T( s )$ are algebraically independent parameters over ${\bf K}$. It is shown that all the invariant polynomials except for the last do not contain the system parameters $( = \,{\text{coefficients of }}\,T ( s ) )$ and that the last invariant polynomial is expressed in terms of the combinatorial canonical form (CCF) of a layered mixed matrix associated with $D( s )$. This implies that the generic dependency of the invariant polynomials on the system parameters can be computed by means of a matroid-theoretic algorithm that involves arithmetic operations in ${\bf K}( s )$.