# Algebraic Theory for Multivariable Linear Systems by Hans Blomberg and Raimo Ylinen (Eds.)

By Hans Blomberg and Raimo Ylinen (Eds.)

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**Extra info for Algebraic Theory for Multivariable Linear Systems**

**Example text**

Example ( 3 ) ) 21 can be transformed to [Lb;;) ................. 01 22 I 2p + 1 0 0 1 P i 2 0 ..................................................................................................................................... e. P@) represents the elementary column operations applied to [A @ ) i -B(p)] to obtain [t@)I 01. Note that here det P@) = 1. It follows that L @ ) given by 24 p:J [2pl+1 is a GCLD of A @ ) and (-) B@). Further we can write 25 [A@)i -B@)] = L @ ) [ A i @ )i -Bib)] with Al(p), B l @ ) left coprime.

4), by S. In single input-single output cases the term “transfer ratio” is occasionally used for (28). The transfer matrix plays an important role in what follows (cf. chapter 6). Note however, that no mapping involving signal spaces is assigned to the transfer matrix at this stage. Properness and strict properness We shall also use the features of the transfer matrix to define the concepts of “properness” and “strict properness”. For every s E C, let %(s) denote the complex matrix obtainable from (28) by substituting s for p in (28).

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.