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Control System Design  Index  Book Contents  Appendix B  Section B.3
B. Smith McMillan Forms
B.3 Smith Form for Polynomial Matrices
Using the above notation, we can manipulate polynomial matrices in
ways that mirror the ways we manipulate matrices of reals. For
example, the following result describes a diagonal form for
polynomial matrices.
Theorem B.1 (Smith form)
Let
be a
polynomial matrix
of rank ;
then
is equivalent to either a matrix
(for )
or to a matrix
(for ), with

(B.3.1) 

(B.3.2) 
where
and
are matrices with all their
elements equal to zero.
Furthermore
are monic polynomials for
,
such that
is a factor in
,
i.e.
divides
.
If ,
then
is equivalent to the square
matrix
.
Proof (by construction)
(i) 

By performing row and column interchange
operations on
,
bring to position (1,1) the least degree
polynomial entry in
.
Say this minimum degree is

(ii) 

Using elementary operation (e03) (see
Definition B.3),
reduce the term in the position (2,1) to degree
.
If the term in position (2,1) becomes zero, then go
to the next step, otherwise, interchange rows 1 and 2 and repeat the
procedure until the term in position (2,1) becomes zero.

(iii) 

Repeat step (ii) with the other elements in the first
column.

(iv) 

Apply the same procedure to all the elements but the
first one in the first row.

(v) 

Go back to step (ii) if nonzero entries due to
step (iv) appear in the first column. Notice that the degree of the entry
(1,1) will fall in each cycle, until we finally end up with a matrix
which can be partitioned as

(B.3.3) 
where
is a monic polynomial.

(vi) 

If there is an element of
which is of
lesser degree than
,
then add the column
where
this element is to the first column and repeat steps (ii) to (v).
Do this until the form (B.3.3) is achieved with
of less or, at most, equal degree to that
of every element in
.
This will yield further
reduction in the degree of the entry in position (1,1).

(vii) 

Make
.

(viii) 

Repeat
the procedure from steps (i) through (viii) to matrix
.

Actually the polynomials
in the above result
can be obtained in a direct fashion, as follows:
(i) 

Compute all minor determinants of
.

(ii) 

Define
as the (monic) greatest common divisor
(g.c.d.) of all
minor determinants of
.
Make
.

(iii) 

Compute the polynomials
as

(B.3.4) 

