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GaussInt

 GIhermite
 Gaussian-integer-only Hermite Normal Form

 Calling Sequence GIhermite(A) GIhermite(A, U)

Parameters

 A - Matrix of Gaussian integers U - name

Description

 • The function GIhermite computes the Hermite Normal Form (reduced row echelon form) of a Matrix of Gaussian integers.
 • The Hermite normal form of A is an upper triangular Matrix H with rank(A) = the number of nonzero rows of H.
 • The Hermite normal form is obtained by doing elementary row operations. This includes interchanging rows, multiplying through a row by a unit in ${Z}_{i}$, and adding an integral multiple of one row to another.
 • One can use transposes to obtain the column form of the Hermite Normal Form.
 • In the case of two arguments, the second argument U will be assigned the transformation Matrix on output, such that the following holds: GIhermite(A) = U . A.

Examples

 > $\mathrm{with}\left(\mathrm{GaussInt}\right):$
 > $A≔\mathrm{Matrix}\left(\left[\left[3-7I,7+11I,11I\right],\left[13-4I,17+12I,19\right]\right]\right)$
 ${A}{:=}\left[\begin{array}{ccc}{3}{-}{7}{}{I}& {7}{+}{11}{}{I}& {11}{}{I}\\ {13}{-}{4}{}{I}& {17}{+}{12}{}{I}& {19}\end{array}\right]$ (1)
 > $B≔\mathrm{GIhermite}\left(A,U\right)$
 ${B}{:=}\left[\begin{array}{ccc}{1}& {-}{59}{-}{2}{}{I}& {-}{82}{-}{8}{}{I}\\ {0}& {198}& {276}{+}{13}{}{I}\end{array}\right]$ (2)
 > $U$
 $\left[\begin{array}{cc}{1}{+}{4}{}{I}& {-}{2}{-}{I}\\ {-}{4}{-}{13}{}{I}& {7}{+}{3}{}{I}\end{array}\right]$ (3)
 > $\mathrm{LinearAlgebra}:-\mathrm{Equal}\left(U\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}A,B\right)$
 ${\mathrm{true}}$ (4)