The command MatrixOverChain(A, rc, R) returns the normal form of A with respect to rc. In broad terms, this is obtained by mapping RegularChains[NormalForm] on the coefficients of A.

The result is viewed as a matrix with coefficients in the total ring of fractions of R/I where I is the saturated ideal of rc.

It is assumed that rc is strongly normalized.

This command is part of the RegularChains[MatrixTools] package, so it can be used in the form MatrixOverChain(..) only after executing the command with(RegularChains[MatrixTools]).  However, it can always be accessed through the long form of the command by using RegularChains[MatrixTools][MatrixOverChain](..).

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$$\mathrm{with}\left(\mathrm{ChainTools}\right)\:$$\mathrm{with}\left(\mathrm{MatrixTools}\right)\:$

$R≔\mathrm{PolynomialRing}\left(\left[x\,y\,z\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$T≔\mathrm{Empty}\left(R\right)\:$

$T≔\mathrm{Chain}\left(\left[\left(z+1\right)\left(z+2\right)\,y^2+z\,\left(x-z\right)\left(x-y\right)\right]\,T\,R\right)$

$T≔\mathrm{regular\_chain}$

$\mathrm{Equations}\left(T\,R\right)$

$\left[x^2+\left(-y-z\right)x+zy\,y^2+z\,z^2+3z+2\right]$

$m≔\mathrm{Matrix}\left(\left[\left[x\,y\,z\right]\,\left[x^2\,y^2\,z^2\right]\,\left[x^3\,y^5\,z^6\right]\right]\right)$

$m≔\left[\begin{array}{ccc}x& y& z\\ x^2& y^2& z^2\\ x^3& y^5& z^6\end{array}\right]$

$\mathrm{MatrixOverChain}\left(m\,T\,R\right)$

$\left[\left[\begin{array}{ccc}x& y& z\\ xy+zx-zy& -z& -3z-2\\ xyz-4zx+3zy-2x+2y-3z-2& -3zy-2y& -63z-62\end{array}\right]\,\mathrm{regular\_chain}\right]$