 GIsmith - Maple Help

GaussInt

 GIsmith
 Gaussian Integer-only Smith Normal Form Calling Sequence GIsmith(A) GIsmith(A, U, V) Parameters

 A - Matrix of Gaussian integers U - name (optional) V - name (optional) Description

 • The function GIsmith computes the Smith normal form S of an n by m Matrix of Gaussian integers.
 • If two n by n Matrices have the same Smith normal form, they are equivalent.
 • The Smith normal form is a diagonal Matrix $S$ where
 $\mathrm{rank}\left(A\right)$ = number of nonzero rows (columns) of $S$
 ${S}_{i,i}$ is in the first quadrant for $0
 ${S}_{i,i}$ divides ${S}_{i+1,i+1}$ for $0
 $\prod _{i=1}^{r}{S}_{i,i}$ divides $\mathrm{det}\left(M\right)$ for all minors $M$ of rank  $0
 • The Smith normal form is obtained by doing elementary row and column operations.  This includes interchanging rows (columns), multiplying through a row (column) by a unit in ${Z}_{i}$, and adding integral multiples of one row (column) to another.
 • In the case of three arguments, the second argument U and the third argument V will be assigned the transformation Matrices on output, such that GIsmith(A) = U . A . V. Examples

 > $\mathrm{with}\left(\mathrm{GaussInt}\right):$
 > $H≔\mathrm{Matrix}\left(\left[\left[-4+7I,8+10I,-6-8I\right],\left[-5+7I,6-6I,5I\right],\left[-10+I,1-3I,-10+5I\right]\right]\right)$
 ${H}{≔}\left[\begin{array}{ccc}{-4}{+}{7}{}{I}& {8}{+}{10}{}{I}& {-6}{-}{8}{}{I}\\ {-5}{+}{7}{}{I}& {6}{-}{6}{}{I}& {5}{}{I}\\ {-10}{+}{I}& {1}{-}{3}{}{I}& {-10}{+}{5}{}{I}\end{array}\right]$ (1)
 > $\mathrm{GIsmith}\left(H\right)$
 $\left[\begin{array}{ccc}{1}& {0}& {0}\\ {0}& {1}& {0}\\ {0}& {0}& {1797}{+}{791}{}{I}\end{array}\right]$ (2)
 > $A≔\mathrm{Matrix}\left(\left[\left[-4-8I,-1-10I,2+3I\right],\left[-1-9I,8+4I,-5+10I\right]\right]\right)$
 ${A}{≔}\left[\begin{array}{ccc}{-4}{-}{8}{}{I}& {-1}{-}{10}{}{I}& {2}{+}{3}{}{I}\\ {-1}{-}{9}{}{I}& {8}{+}{4}{}{I}& {-5}{+}{10}{}{I}\end{array}\right]$ (3)
 > $B≔\mathrm{GIsmith}\left(A,U,V\right)$
 ${B}{≔}\left[\begin{array}{ccc}{1}& {0}& {0}\\ {0}& {1}& {0}\end{array}\right]$ (4)
 > $U$
 $\left[\begin{array}{cc}{-1}{+}{4}{}{I}& {-1}{-}{I}\\ {5}{-}{10}{}{I}& {2}{+}{3}{}{I}\end{array}\right]$ (5)
 > $V$
 $\left[\begin{array}{ccc}{0}& {43}{+}{30}{}{I}& {101}{+}{8}{}{I}\\ {0}& {-28}{-}{29}{}{I}& {-75}{-}{21}{}{I}\\ {1}& {66}{-}{21}{}{I}& {89}{-}{99}{}{I}\end{array}\right]$ (6)
 > $\mathrm{LinearAlgebra}:-\mathrm{Equal}\left(U·A·V,B\right)$
 ${\mathrm{true}}$ (7)