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LinearAlgebra[Modular][Rank] - compute the rank of a mod m Matrix
LinearAlgebra[Modular][RankProfile] - compute the rank profile of a square mod m Matrix
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Calling Sequence
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Rank(m, A, meth)
RankProfile(m, A, meth)
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Parameters
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m
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modulus
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A
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mod m Matrix
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meth
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(optional) keyword for choice of method
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Description
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The Rank function returns the rank of the input mod m Matrix, while the RankProfile function returns a list of 'rank' elements describing the rank profile of the input mod m Matrix.
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The rank profile list is simply a list of the location of the first non-zero entry in each nontrivial row in the row reduced form of the Matrix.
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The following methods are available:
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REF
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(default) Compute using standard row-reduction (Row Echelon Form)
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inplaceREF
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Compute using standard row-reduction in-place in the input Matrix
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RET
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Compute using a Row Echelon Transformation approach
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inplaceRET
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Compute using a Row Echelon Transformation in-place in the input Matrix
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Note that the two inplace methods available will destroy the data in the input Matrix, while the other two methods will generate a copy of the Matrix in which to perform the computation.
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The RET methods are likely to be faster for large matrices, but may fail if the modulus is composite.
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These commands are part of the LinearAlgebra[Modular] package, so they can be used in the form Rank(..) and RankProfile(..) only after executing the command with(LinearAlgebra[Modular]). However, they can always be used in the form LinearAlgebra[Modular][Rank](..) and LinearAlgebra[Modular][RankProfile](..).
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Examples
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![with(LinearAlgebra[Modular])](/support/helpjp/helpview.aspx?si=4523/file06085/math142.png)
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![M := Mod(p, [[2, 11, 1, 80], [31, 16, 27, 36], [8, 32, 32, 31], [25, 4, 90, 63]], integer[])](/support/helpjp/helpview.aspx?si=4523/file06085/math153.png)
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![M := Matrix([[2, 11, 1, 80], [31, 16, 27, 36], [8, 32, 32, 31], [25, 4, 90, 63]])](/support/helpjp/helpview.aspx?si=4523/file06085/math156.png)
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![4, [1, 2, 3, 4]](/support/helpjp/helpview.aspx?si=4523/file06085/math163.png)
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With an inplace method the input Matrix is altered
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![4, Matrix([[1, 54, 49, 40], [0, 1, 58, 26], [0, 0, 1, 35], [0, 0, 0, 1]])](/support/helpjp/helpview.aspx?si=4523/file06085/math174.png)
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And a case that is not full rank
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![M := Mod(p, [[2, 2, 1, 80], [31, 31, 27, 36], [8, 8, 32, 31], [25, 25, 90, 63]], integer[])](/support/helpjp/helpview.aspx?si=4523/file06085/math180.png)
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![M := Matrix([[2, 2, 1, 80], [31, 31, 27, 36], [8, 8, 32, 31], [25, 25, 90, 63]])](/support/helpjp/helpview.aspx?si=4523/file06085/math183.png)
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![[1, 3, 4]](/support/helpjp/helpview.aspx?si=4523/file06085/math190.png)
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A composite example where the RET method is unsuccessful
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![M := Matrix([[3, 2], [2, 3]], datatype = integer)](/support/helpjp/helpview.aspx?si=4523/file06085/math203.png)
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![M := Matrix([[3, 2], [2, 3]])](/support/helpjp/helpview.aspx?si=4523/file06085/math206.png)
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| (9) |
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Note that this is only because this is a case where the row echelon form exists, but the row echelon transform cannot be written in the required form.
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Download Help Document
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