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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 $\mathbf{\ensuremath{\boldsymbol{\Pi}} } (s)$ be a $m_1\times m_2$ polynomial matrix of rank $r$; then $\mathbf{\ensuremath{\boldsymbol{\Pi}} } (s)$ is equivalent to either a matrix $\mathbf{\ensuremath{\boldsymbol{\Pi}} }_f (s)$ (for $m_1<m_2$) or to a matrix $\mathbf{\ensuremath{\boldsymbol{\Pi}} }_c (s)$ (for $m_2<m_1$), with


\begin{displaymath}\mathbf{\ensuremath{\boldsymbol{\Pi}} }_f (s)= \begin{bmatrix... ...} }_c (s)=\begin{bmatrix}\mathbf{E}(s)\\ \Theta_c\end{bmatrix}\end{displaymath} (B.3.1)


\begin{displaymath}\mathbf{E}(s)=\diag(\overline{\epsilon}_1(s),\ldots,\overline{\epsilon}_r(s),0,\dots,0) \end{displaymath} (B.3.2)

where $\Theta_f$ and $\Theta_c$ are matrices with all their elements equal to zero.

Furthermore $\overline{\epsilon}_i(s)$ are monic polynomials for $i=1,2,\ldots,r$, such that $\overline{\epsilon}_i(s)$ is a factor in $\overline{\epsilon}_{i+1}(s)$, i.e. $\overline{\epsilon}_i(s)$ divides $\overline{\epsilon}_{i+1}(s)$.

If $m_1=m_2$, then $\mathbf{\ensuremath{\boldsymbol{\Pi}} } (s)$ is equivalent to the square matrix $\mathbf{E}(s)$.

Proof (by construction)

(i) By performing row and column interchange operations on $\mathbf{\ensuremath{\boldsymbol{\Pi}} } (s)$, bring to position (1,1) the least degree polynomial entry in $\mathbf{\ensuremath{\boldsymbol{\Pi}} } (s)$. Say this minimum degree is $\nu_1$

(ii) Using elementary operation (e03) (see Definition B.3), reduce the term in the position (2,1) to degree $\nu_2<\nu_1$. 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


 \begin{displaymath} \ensuremath{\mathbf{\ensuremath{\boldsymbol{\Pi}}}(s)} = \... ...\Pi}}_j}(s)} & & \\ 0& & & & & \\ 0& & & & & \\ \end{bmatrix}\end{displaymath} (B.3.3)

where $\overline{\pi}_{11}^{(j)}(s)$ is a monic polynomial.

(vi) If there is an element of $\ensuremath{\mathbf{\ensuremath{\boldsymbol{\Pi}}_j}(s)} $ which is of lesser degree than $\overline{\pi}_{11}^{(j)}(s)$, 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 $\overline{\pi}_{11}^{(j)}(s)$ of less or, at most, equal degree to that of every element in $\ensuremath{\mathbf{\ensuremath{\boldsymbol{\Pi}}_j}(s)} $. This will yield further reduction in the degree of the entry in position (1,1).

(vii) Make $\overline{\epsilon}_1(s)=\overline{\pi}_{11}^{(j)}(s)$.

(viii) Repeat the procedure from steps (i) through (viii) to matrix $\ensuremath{\mathbf{\ensuremath{\boldsymbol{\Pi}}_j}(s)} $.

Actually the polynomials $\overline{\epsilon}_i(s)$ in the above result can be obtained in a direct fashion, as follows:

(i) Compute all minor determinants of $\mathbf{\ensuremath{\boldsymbol{\Pi}} } (s)$.

(ii) Define $\chi_i(s)$ as the (monic) greatest common divisor (g.c.d.) of all $i\times i$ minor determinants of $\mathbf{\ensuremath{\boldsymbol{\Pi}} } (s)$. Make $\chi_0(s)=1$.

(iii) Compute the polynomials $\overline{\epsilon}_i(s)$ as


\begin{displaymath}\overline{\epsilon}_i(s)=\frac{\chi_i(s)}{\chi_{i-1}(s)} \end{displaymath} (B.3.4)