MatrixOverChain - Maple Help

RegularChains[MatrixTools]

 MatrixOverChain
 normal form of a matrix with respect to a regular chain

 Calling Sequence MatrixOverChain(A, rc, R)

Parameters

 A - Matrix with coefficients in the field of fractions of R rc - regular chain of R R - polynomial ring

Description

 • 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](..).

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$$\mathrm{with}\left(\mathrm{ChainTools}\right):$$\mathrm{with}\left(\mathrm{MatrixTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,z\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $T≔\mathrm{Empty}\left(R\right):$
 > $T≔\mathrm{Chain}\left(\left[\left(z+1\right)\left(z+2\right),{y}^{2}+z,\left(x-z\right)\left(x-y\right)\right],T,R\right)$
 ${T}{≔}{\mathrm{regular_chain}}$ (2)
 > $\mathrm{Equations}\left(T,R\right)$
 $\left[{{x}}^{{2}}{+}\left({-}{y}{-}{z}\right){}{x}{+}{z}{}{y}{,}{{y}}^{{2}}{+}{z}{,}{{z}}^{{2}}{+}{3}{}{z}{+}{2}\right]$ (3)
 > $m≔\mathrm{Matrix}\left(\left[\left[x,y,z\right],\left[{x}^{2},{y}^{2},{z}^{2}\right],\left[{x}^{3},{y}^{5},{z}^{6}\right]\right]\right)$
 ${m}{≔}\left[\begin{array}{ccc}{x}& {y}& {z}\\ {{x}}^{{2}}& {{y}}^{{2}}& {{z}}^{{2}}\\ {{x}}^{{3}}& {{y}}^{{5}}& {{z}}^{{6}}\end{array}\right]$ (4)
 > $\mathrm{MatrixOverChain}\left(m,T,R\right)$
 $\left[\left[\begin{array}{ccc}{x}& {y}& {z}\\ {x}{}{y}{+}{z}{}{x}{-}{z}{}{y}& {-}{z}& {-}{3}{}{z}{-}{2}\\ {x}{}{y}{}{z}{-}{4}{}{z}{}{x}{+}{3}{}{z}{}{y}{-}{2}{}{x}{+}{2}{}{y}{-}{3}{}{z}{-}{2}& {-}{3}{}{z}{}{y}{-}{2}{}{y}& {-}{63}{}{z}{-}{62}\end{array}\right]{,}{\mathrm{regular_chain}}\right]$ (5)