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## B. Smith-McMillan Forms

### B.4 Smith-McMillan Form for Rational Matrices

A straightforward application of Theorem B.1 leads to the following result, which gives a diagonal form for a rational transfer-function matrix:

Theorem 2.2 (Smith-McMillan form)
Let be an matrix transfer function, where are rational scalar transfer functions:

 (B.4.1)

where is an polynomial matrix of rank and is the least common multiple of the denominators of all elements .

Then, is equivalent to a matrix , with

 (B.4.2)

where is a pair of monic and coprime polynomials for .

Furthermore, is a factor of and is a factor of .

##### Proof

We write the transfer-function matrix as in (B.4.1). We then perform the algorithm outlined in Theorem B.1 to convert to Smith normal form. Finally, canceling terms for the denominator leads to the form given in (B.4.2).

We use the symbol to denote , which is the Smith-McMillan form of the transfer-function matrix .

We illustrate the formula of the Smith-McMillan form by a simple example.

Example B.1 Consider the following transfer-function matrix

 (B.4.3)

We can then express in the form (B.4.1):

 (B.4.4)

The polynomial matrix can be reduced to the Smith form defined in Theorem B.1. To do that, we first compute its greatest common divisors:

 (B.4.5)

 (B.4.6)

 (B.4.7)