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RegularChains[MatrixTools]

 LowerEchelonForm
 lower echelon form of a matrix modulo a regular chain

 Calling Sequence LowerEchelonForm(A, rc, R)

Parameters

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

Description

 • The command LowerEchelonForm(A, rc, R) returns a list of pairs $\left[{B}_{i},{\mathrm{rc}}_{i}\right]$ where ${\mathrm{rc}}_{i}$ is a regular chain, and ${B}_{i}$ is the lower echelon form of A modulo the saturated ideal of rc_i.
 • All the returned regular chains ${\mathrm{rc}}_{i}$ form a triangular decomposition of rc (in the sense of Kalkbrener).
 • It is assumed that rc is strongly normalized.
 • The algorithm is an adaptation of the algorithm of Bareiss.
 • This command is part of the RegularChains[MatrixTools] package, so it can be used in the form LowerEchelonForm(..) 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][LowerEchelonForm](..).

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)
 > $m≔\mathrm{Matrix}\left(\left[\left[x,y,z\right],\left[x+1,y+2,z+3\right],\left[x+4,y+5,z+6\right]\right]\right)$
 ${m}{:=}\left[\begin{array}{ccc}{x}& {y}& {z}\\ {x}{+}{1}& {y}{+}{2}& {z}{+}{3}\\ {x}{+}{4}& {y}{+}{5}& {z}{+}{6}\end{array}\right]$ (4)
 > $\mathrm{lem}≔\mathrm{LowerEchelonForm}\left(m,T,R\right)$
 ${\mathrm{lem}}{:=}\left[\left[\left[\begin{array}{ccc}{6}& {0}& {0}\\ {0}& {3}& {0}\\ {x}{+}{4}& {y}{+}{5}& {z}{+}{6}\end{array}\right]{,}{\mathrm{regular_chain}}\right]{,}\left[\left[\begin{array}{ccc}{12}& {0}& {0}\\ {-}{6}& {3}& {0}\\ {x}{+}{4}& {y}{+}{5}& {z}{+}{6}\end{array}\right]{,}{\mathrm{regular_chain}}\right]{,}\left[\left[\begin{array}{ccc}{0}& {0}& {0}\\ {-}{6}& {-}{3}& {0}\\ {x}{+}{4}& {y}{+}{5}& {z}{+}{6}\end{array}\right]{,}{\mathrm{regular_chain}}\right]{,}\left[\left[\begin{array}{ccc}{-}{3}{}{x}{+}{6}{}{y}{+}{6}& {0}& {0}\\ {3}{}{x}& {3}{}{y}{+}{3}& {0}\\ {x}{+}{4}& {y}{+}{5}& {z}{+}{6}\end{array}\right]{,}{\mathrm{regular_chain}}\right]\right]$ (5)