** suivant:** Isometries
** monter:** Matrix reduction
** précédent:** Hermite normal form :
** Table des matières**
** Index**

##

Smith normal form : `ismith`

`ismith` takes as argument a matrix with coefficients in
.

`ismith` returns three matrices `U,B` and `V` such
that `B=U*A*V`, `U` and `V` are invertible in ,
`B` is diagonal, and `B[i,i]` divides `B[i+1,i+1]`.
The coefficients `B[i,i]` are called
invariant factors, they are used to describe
the structure of finite abelian groups.

Input :
`A:=[[9,-36,30],[-36,192,-180],[30,-180,180]];
U,B,V:=ismith(A)`

Output :
`[[-3,0,1],[6,4,3],[20,15,12]],
[[3,0,0],[0,12,0],[0,0,60]],
[[1,24,-30],[0,1,0],[0,0,1]] `

The invariant factors are 3, 12 and 60.

giac documentation written by Renée De Graeve and Bernard Parisse