For an instance $({\bm{S}}, {\bm{g}}, {\bm{N}})$ of \[prob:sim\_pade\], in the context of defining matrices, we will be using ${\bm{S}}$ and ${\bm{g}}$ as vectors, and by ${  \ifx\null\null  \Gamma_{{\bm{g}}}  \else  \Gamma_{{\bm{g}}_{\null}}  \fi}$ denote the diagonal matrix with the entries of ${\bm{g}}$ on its diagonal.

Row reduced
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Although row reducedness can be defined for matrices of arbitrary shape and rank, it suffices here to consider the case of matrices of full row rank. A matrix $R \in {{\mathsf{K}}}[x]^{n \times m}$ is [*row reduced*]{} if ${\rm LM}(R)$ has full row rank, and [*${\bm{s}}$-row reduced*]{} if ${\rm LM}_{{\bm{s}}}(R)$ has full row rank. Every $A \in {{\mathsf{K}}}[x]^{n \times m}$ of full row rank is left equivalent to a matrix $R \in {{\mathsf{K}}}[x]^{n \times m}$ that is ${{\bm{s}}}$-row reduced. The rows of $R$ give a basis for ${{\textnormal{Row}}}(A)$ that is minimal in the following sense: the list of ${{\bm{s}}}$-degrees of the rows of $R$, when sorted in non-decreasing order, will be lexicographically minimal. An important feature of row reduced matrices is the so-called “predictable degree”-property [@kailath_linear_1980 Theorem 6.3-13]: for any ${\bm{v}} \in {{\mathsf{K}}}[x]^{1 \times n}$, we have $$\deg_{{\bm{s}}}({\bm{v}} R) = \max_{i=1,\ldots,n}( \deg_{{\bm{s}}} {\rm row}(R,i)
+ \deg v_i ) \ .$$

A canonical ${\bm{s}}$-reduced basis is 