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B. Smith McMillan Forms

B.2 Polynomial Matrices

Multivariable transfer functions depend on polynomial matrices. There are a number of related terms that are used. Some of these are introduced here:

Definition B.1 A matrix $\ensuremath{\mathbf{\ensuremath{\boldsymbol{\Pi}}}(s)} =[p_{ik}(s)]\in {\mathbb{R} }^{n_1\times n_2}$ is a polynomial matrix  if $p_{ik}(s)$ is a polynomial in $s$, for $i=1,2,\ldots,n_1$ and $k=1,2,\ldots,n_2$.


Definition B.2 A polynomial matrix $\ensuremath{\boldsymbol{\Pi}} (s)$ is said to be a unimodular matrix  if its determinant is a constant. Clearly, the inverse of a unimodular matrix is also a unimodular matrix.


Definition B.3 An elementary operation on a polynomial matrix is one of the following three operations:

(eo1)
interchange of two rows or two columns;
(eo2)
multiplication of one row or one column by a constant;
(eo3)
addition of one row (column) to another row (column) times a polynomial.


Definition B.4 A left (right) elementary matrix is a matrix such that, when it multiplies from the left (right) a polynomial matrix, then it performs a row (column) elementary operation on the polynomial matrix. All elementary matrices are unimodular.


Definition B.5 Two polynomial matrices $\mathbf{\ensuremath{\boldsymbol{\Pi}} }_1(s)$ and $\mathbf{\ensuremath{\boldsymbol{\Pi}} }_2(s)$ are equivalent matrices, if there exist sets of left and right elementary matrices, $\{\mathbf{L}_1(s),\mathbf{L}_2(s),\ldots,\mathbf{L}_{k1}\}$ and $\{\mathbf{R}_1(s), \mathbf{R}_2(s), \ldots, \mathbf{R}_{k2}\}$, respectively, such that
\begin{displaymath}\mathbf{\ensuremath{\boldsymbol{\Pi}} }_1(s)=\mathbf{L}_{k1}(... ...}} }_2(s)\mathbf{R}_1(s)\mathbf{R}_2(s)\cdots \mathbf{R}_{k2} \end{displaymath} (B.2.1)


Definition B.6 The rank of a polynomial matrix is the rank of the matrix almost everywhere in $s$. The definition implies that the rank of a polynomial matrix is independent of the argument.


Definition B.7 Two polynomial matrices $\ensuremath{\mathbf{V}(s)} $ and $\ensuremath{\mathbf{W}(s)} $ having the same number of columns (rows) are right (left) coprime if all common right (left) factors are unimodular matrices.


Definition B.8 The degree $\partial_{ck}$ ( $\partial_{rk}$) of the $k^{th}$column (row) $[\ensuremath{\mathbf{V}(s)} ]_{*k}$ ( $[\ensuremath{\mathbf{V}(s)} ]_{k*}$) of a polynomial matrix $\ensuremath{\mathbf{V}(s)} $ is the degree of highest power of $s$ in that column (row).


Definition B.9 A polynomial matrix $\ensuremath{\mathbf{V}(s)}\in \mathbb{C} ^{m\times m} $ is column proper if

\begin{displaymath}\lim_{s\rightarrow \infty}\det(\ensuremath{\mathbf{V}(s)}\dia... ...}}, s^{- \partial_{c2}}, \ldots, s^{- \partial_{cm}} \right)) \end{displaymath} (B.2.2)

has a finite, nonzero value.


Definition B.10 A polynomial matrix $\ensuremath{\mathbf{V}(s)}\in \mathbb{C} ^{m\times m} $ is row proper if

\begin{displaymath}\lim_{s\rightarrow \infty}\det(\diag\left(s^{- \partial_{r1}}... ...dots, s^{- \partial_{rm}} \right)\ensuremath{\mathbf{V}(s)} ) \end{displaymath} (B.2.3)

has a finite, nonzero value.