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RegularChains[MatrixTools]

 IsZeroMatrix
 check whether a matrix is null modulo a regular chain

 Calling Sequence IsZeroMatrix(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 IsZeroMatrix(A, rc, R) returns true if and only if A is null modulo 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 IsZeroMatrix(..) 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][IsZeroMatrix](..).

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$$\mathrm{with}\left(\mathrm{ChainTools}\right):$$\mathrm{with}\left(\mathrm{MatrixTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[y,z\right]\right)$
 ${R}{:=}{\mathrm{polynomial_ring}}$ (1)
 > $\mathrm{rc}≔\mathrm{Empty}\left(R\right)$
 ${\mathrm{rc}}{:=}{\mathrm{regular_chain}}$ (2)
 > $\mathrm{rc}≔\mathrm{Chain}\left(\left[{z}^{4}+1,{y}^{2}-{z}^{2}\right],\mathrm{rc},R\right):$
 > $\mathrm{Equations}\left(\mathrm{rc},R\right)$
 $\left[{{y}}^{{2}}{-}{{z}}^{{2}}{,}{{z}}^{{4}}{+}{1}\right]$ (3)
 > $m≔\mathrm{Matrix}\left(\left[\left[1,y+z\right],\left[0,y-z\right]\right]\right)$
 ${m}{:=}\left[\begin{array}{cc}{1}& {y}{+}{z}\\ {0}& {y}{-}{z}\end{array}\right]$ (4)
 > $\mathrm{IsZeroMatrix}\left(m,\mathrm{rc},R\right)$
 ${\mathrm{false}}$ (5)