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common denominator for the entries Gij (s)of G(s). Then, G(s) can be written as N1(s) 1n(s) G(s)=dsN(s)=ds where N(s)is a poly nomial matrix (not a transfer matrix). There are three element ary types of row or column operations to perform on N(s) Interchange of two rows or columns Multiplication of a row or a column with a const ant Addition of one row or column multiplied with a polynomial to another A common property of these operations is, that they do not change the rank of the matrix ( s). Each of these elementary operations can be represented as a pre- or post multiplication of N(s by a suit able matrix L(s) called an elementary matric. It can be shown, that all element ary matrices are unimodular. Now, N(s can be rewritten as a sequen ce of row and column operations (8)=L1(s)S(s)L2(8) Ln(s)diag{∈1(s),E2(8),……,6r(s),0,0,…,0}L2(s) (4.12) Here, S(s is a pseudo-diagonal polynomial matrix. S(s) is called the Smith form of N(s The polynomials Ei(s) are monic and have the following divisibility properties 1 (4.13) Hence, the Smith form of a polynomial matrix is equivalent to the Smith-McMillan form of a transfer matrix. Now, the point is that the polynomials Ei(s) can be determined from the determinant divisors (4.14) Di()=greatest common divisor for all i x i sub determinants of N(s) (4.15) where every of the greatest common divisors are normalized to a monic polynomial. It can be shown, see [Mac89, Pages 40-43 that the polynomials Ei() are given by (4.16) Hence, the Smith-McMillan form of G(s) is given by M()=(s) 17) In summary, this est ablishes the following procedure for determining the Smith-McMillan form of a transfer matrix G(s) ' "#5! % %! ' & 6 6 + % 7 % ' "##! "#(! ) % ' "#,! ) % && % ' "#"! "#/! 8&23 . "59",: ' "#-! ) && ' "#1! && % '