The Algebra of Polynomial Matrices

Definition.    Polynomial matrix
               Determinant
               Inverse      (Recall some lin. alg.)
               Unimodular (square) Matrix

      Factorizations: N(s) D(s)^{-1}
          non-uniqueness
          * Right coprime & irreducible factorization
                            Minimal factorization

     Theorem.   1) Irreducible \equiv Minimal
                2) Degree of det of denominator of
                    minimal factorization is well-defined
                3) Def:  This is MacMillan degree

 Recall:   Euclidean algorithm for polynomials

  Theorem  on Smith-MacMillan Normal Form & Invariant Factors

   Proof is a polynomial version of Gaussian Elimination

    1) Get matrix in diagonal form
    2) Explain how minors change under a row operation; this
        implies that the gcd of the kxk minors does not change, 
        and hence we deduce the fact anout invariant factors