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RegularChains[MatrixTools][MatrixOverChain] - normal form of a matrix with respect to a regular chain
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Calling Sequence
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MatrixOverChain(A, rc, R)
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Parameters
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A
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Matrix with coefficients in the field of fractions of R
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rc
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regular chain of R
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R
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polynomial ring
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Description
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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.
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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.
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It is assumed that rc is strongly normalized.
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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](..).
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Examples
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![R := PolynomialRing([x, y, z])](/support/helpjp/helpview.aspx?si=4971/file06508/math107.png)
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![T := Chain([(z+1)*(z+2), y^2+z, (x-z)*(x-y)], T, R)](/support/helpjp/helpview.aspx?si=4971/file06508/math118.png)
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| (2) |
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![[x^2+(-y-z)*x+z*y, y^2+z, z^2+3*z+2]](/support/helpjp/helpview.aspx?si=4971/file06508/math128.png)
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![m := Matrix([[x, y, z], [x^2, y^2, z^2], [x^3, y^5, z^6]])](/support/helpjp/helpview.aspx?si=4971/file06508/math132.png)
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![m := Matrix([[x, y, z], [x^2, y^2, z^2], [x^3, y^5, z^6]])](/support/helpjp/helpview.aspx?si=4971/file06508/math135.png)
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![[Matrix([[x, y, z], [x*y+z*x-z*y, -z, -2-3*z], [x*z*y-4*z*x-2*x+2*y-2+3*z*y-3*z, -2*y-3*z*y, -63*z-62]]), regular_chain]](/support/helpjp/helpview.aspx?si=4971/file06508/math142.png)
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See Also
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Chain, Empty, Equations, IsStronglyNormalized, IsZeroMatrix, JacobianMatrix, LowerEchelonForm, Matrix, MatrixInverse, MatrixMultiply, MatrixTools, NormalForm, PolynomialRing, RegularChains
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