Applicable Algebraic Geometry

28 September 2000

Frank Sottile

Outline for Lecture:

Finish Proof of Smith Normal Form

  * Inductive Claim & Proof
  * Claim about invariant factors
    - Recall Cauchy-Binet formula
    - Maple Handout of computation
    - Complete proof.

3 Equivalent conditions for coprime matrices

Minimal factorization is equivalent to irreducible

Remark on Algorithm to compute minimal factorizations

Exercise:  Write a Maple, Mathematica,... script to 
            run this algorithm.

   Goal: compute minimal factorizations for transfer functions
         C ( sI_n - A )^(-1) B      or
         H ( sI_q - F )^(-1) G + K

We do not complete our tour of polynomial matrices - this
 will be done in the appendix/handout I am writing.

---------------------------------------------------------
Recall:
   Feedback Control using a dynamic compensator,
   Closed loop system,
   Form of the charactreristic equation.

Fact: A, B, C, controllable & abservable imply that 
      a minimal factorisation
         C ( sI_n - A )^(-1) B  = N(s) D(s)^(-1)  
      has
          det (sI_n-A) = det D(s).

(Give Heuristic!)

In case of static control and distinct eigrnvalues, give geometric
conditions for pole placement problem