Gausselim - Maple Help

Gausselim

inert Gaussian elimination

Gaussjord

inert Gauss Jordan elimination

 Calling Sequence Gausselim(A) mod p Gaussjord(A) mod p Gausselim(A, 'r', 'd') mod p Gaussjord(A, 'r', 'd') mod p

Parameters

 A - Matrix 'r' - (optional) for returning the rank of A 'd' - (optional) for returning the determinant of A 'p' - an integer, the modulus

Description

 • The Gausselim and Gaussjord functions are placeholders for representing row echelon forms of the rectangular matrix A.
 • The commands Gausselim(A,...) mod p and Gassjord(A,...) mod p apply Gaussian elimination with row pivoting to A, a rectangular matrix over a finite ring of characteristic p. This includes finite fields, GF(p), the integers mod p, and GF(p^k) where elements of GF(p^k) are expressed as polynomials in RootOfs.
 • The result of the Gausselim command is a an upper triangular matrix B in row echelon form.  The result of the Gaussjord command is also an upper triangular matrix B but in reduced row echelon form.
 • If an optional second parameter is specified, and it is a name, it is assigned the rank of the matrix A.
 • If A is an $m$ by $n$ matrix with $m\le n$ and if an optional third parameter is also specified, and it is a name, it is assigned the determinant of the matrix A[1..m,1..m].

Examples

 > $A≔\mathrm{Matrix}\left(\left[\left[1,2,3\right],\left[1,3,0\right],\left[1,4,3\right]\right]\right)$
 ${A}{≔}\left[\begin{array}{ccc}{1}& {2}& {3}\\ {1}& {3}& {0}\\ {1}& {4}& {3}\end{array}\right]$ (1)
 > $\mathrm{Gausselim}\left(A\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}5$
 $\left[\begin{array}{ccc}{1}& {2}& {3}\\ {0}& {1}& {2}\\ {0}& {0}& {1}\end{array}\right]$ (2)
 > $B≔\mathrm{ArrayTools}\left[\mathrm{Concatenate}\right]\left(2,A,\mathrm{LinearAlgebra}\left[\mathrm{IdentityMatrix}\right]\left(3\right)\right)$
 ${B}{≔}\left[\begin{array}{cccccc}{1}& {2}& {3}& {1}& {0}& {0}\\ {1}& {3}& {0}& {0}& {1}& {0}\\ {1}& {4}& {3}& {0}& {0}& {1}\end{array}\right]$ (3)
 > $\mathrm{Gaussjord}\left(B\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}5$
 $\left[\begin{array}{cccccc}{1}& {0}& {0}& {4}& {1}& {1}\\ {0}& {1}& {0}& {2}& {0}& {3}\\ {0}& {0}& {1}& {1}& {3}& {1}\end{array}\right]$ (4)
 > $\mathrm{Inverse}\left(A\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}5$
 $\left[\begin{array}{ccc}{4}& {1}& {1}\\ {2}& {0}& {3}\\ {1}& {3}& {1}\end{array}\right]$ (5)
 > $\mathrm{alias}\left(a=\mathrm{RootOf}\left({x}^{4}+x+1\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}2\right):$
 > $A≔\mathrm{Matrix}\left(\left[\left[1,a,{a}^{2}\right],\left[a,{a}^{2},{a}^{3}\right],\left[{a}^{2},{a}^{3},1\right]\right]\right)$
 ${A}{≔}\left[\begin{array}{ccc}{1}& {a}& {{a}}^{{2}}\\ {a}& {{a}}^{{2}}& {{a}}^{{3}}\\ {{a}}^{{2}}& {{a}}^{{3}}& {1}\end{array}\right]$ (6)
 > $\mathrm{Gausselim}\left(A,'r','d'\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}2$
 $\left[\begin{array}{ccc}{1}& {a}& {{a}}^{{2}}\\ {0}& {0}& {a}\\ {0}& {0}& {0}\end{array}\right]$ (7)
 > $r$
 ${2}$ (8)
 > $d$
 ${0}$ (9)