Smith normal form in ℤ

15.2.14 Smith normal form in ℤ

A matrix B is in Smith normal form if the only non-zero entries are on the diagonal (for non-square matrices, this simply means that b_ij=0 for i≠j) and b_i,i divides b_i+1,i+1. The elements b_i,i are called invariant factors and are used to describe the structure of finite abelian groups.

For any matrix A with coefficients in ℤ, there exist matrices U and V, invertible in ℤ, such that B=U A V is in Smith normal form and has coefficients in ℤ. The ismith command finds the matrices U, B and V.

ismith takes A, a matrix with coefficients in ℤ.
ismith(A) returns a list [U,B,V] of three matrices such that B=UAV is in Smith normal form and U and V are invertible in ℤ.

Example

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

⎡
⎢
⎢
⎣

−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.