By Hans Blomberg and Raimo Ylinen (Eds.)
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Additional resources for Algebraic Theory for Multivariable Linear Systems
________. , where Al(p) is of CUT-form (Al(p) square, det Al(p) # 0) coincide. Hence such matrices can also be said to be of CUT-form. Let P ( p ) be a unimodular matrix and let B(p) be a suitable polynomial matrix so that P(p)B(p) is of CUT-form. Then P ( p ) is unique if P(p)B(p) is of the form (14), but nonunique if P(p)B(p) is of the form (15) with one or more zero rows. 3 triangular form. Any unimodular matrix is column equivalent to the identity matrix. To avoid ambiguity, we shall use the CUT- and CLT-notations only for canonical forms with respect to row equivalence.
12 4. Example. The CUT-form of A @ ) as given by (4)is (cf. ________. , where Al(p) is of CUT-form (Al(p) square, det Al(p) # 0) coincide. Hence such matrices can also be said to be of CUT-form. Let P ( p ) be a unimodular matrix and let B(p) be a suitable polynomial matrix so that P(p)B(p) is of CUT-form. Then P ( p ) is unique if P(p)B(p) is of the form (14), but nonunique if P(p)B(p) is of the form (15) with one or more zero rows. 3 triangular form. Any unimodular matrix is column equivalent to the identity matrix.
Note that these operations do not include multiplication of a row by a nonconstant polynomial. e. by matrices that are obtained by applying the elementary row operations to the identity matrix Z of the proper size. The elementary matrices and their products are unimodular. Now the determinant of a triangular matrix is equal to the product of the diagonal entries. A unimodular polynomial matrix must therefore be row equivalent to the identity matrix. It then follows that every unimodular matrix can be expressed as a product of elementary matrices and these matrices can be found by triangularization of the matrix under consideration.
Algebraic Theory for Multivariable Linear Systems by Hans Blomberg and Raimo Ylinen (Eds.)