Coprime factorization of rational matrix over the polynomial ring

Question

Dear esteemed colleagues,

I have a question regarding the coprime factorization of rational matrices over the polynomial ring. Consider the following rational matrix

$$ R(s) = \begin{pmatrix} \frac{1}{s+1} \\ \frac{1}{s+1} \end{pmatrix}, $$

for which I know a right coprime factorization over the polynomial ring, i.e., $R(s) = N_r(s) D_r^{-1} (s) = \begin{pmatrix} 1 \\ 1 \end{pmatrix} (s+1)^{-1}$. However, I'm interested in a left coprime factorization, i.e., $R(s) = D_\ell^{-1} (s) N_\ell(s) $. I have found the following factorization

$$ D_\ell = (s+1) I_2, \quad \quad N_\ell = \begin{pmatrix} 1 \\ 1 \end{pmatrix}, $$

but it is unfortunately not coprime.

Does there exist a leftcoprime factorization over the polynomial ring for this example? Does a left AND right coprime factorization over the polynomial ring always exist for general non-square rational matrices?

Thanks.

Answer 1

I think the Smith-McMillan from will lead to a left coprime factorization, but by observation, we could have $$ \begin{bmatrix} s+1&s+1\\ 1&-1 \end{bmatrix}^{-1} \begin{bmatrix}2\\0 \end{bmatrix}= \begin{bmatrix} \frac{1}{s+1}\\\frac{1}{s+1} \end{bmatrix} $$