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15.2.15  Smith normal form

The smith command finds the Smith normal form of a matrix with elements in a field K.

Examples

M:=([[5,-2,3,6],[1,-3,1,3],[7,-6,-4,7],[-2,-4,-3,0]]) % 17:; A:=x*idn(4)-M
     
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
x+⎛
⎝		−5⎞
⎠		%17
2%17
⎛
⎝		−3⎞
⎠		%17
⎛
⎝		−6⎞
⎠		%17
⎛
⎝		−1⎞
⎠		%17
x+3%17
⎛
⎝		−1⎞
⎠		%17
⎛
⎝		−3⎞
⎠		%17
⎛
⎝		−7⎞
⎠		%17
6%17x+4%17
⎛
⎝		−7⎞
⎠		%17
2%174%173%17x
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
          
U,D,V:=smith(A):;
     
 
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0%17
⎛
⎝		−1⎞
⎠		%17
0%170%17
0%170%176%174%17
⎛
⎝		−2 x+5⎞
⎠		%17
⎛
⎝		−4 x−5⎞
⎠		%17
⎛
⎝		−3 x−6⎞
⎠		%17
⎛
⎝		x2−3 x+6⎞
⎠		%17
⎛
⎝		2 x2+5 x+6⎞
⎠		%17
⎛
⎝		4 x2+8 x+2⎞
⎠		%17
⎛
⎝		3 x2+4 x+1⎞
⎠		%17
⎛
⎝		−x3−2 x2+2 x−6⎞
⎠		%17
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
         
 
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1%17
⎛
⎝		x+3⎞
⎠		%17
⎛
⎝		−6 x2−3 x−7⎞
⎠		%17
⎛
⎝		6 x5+2 x4−2 x3+x2−8 x+6⎞
⎠		%17
0%171%17
⎛
⎝		−6 x−2⎞
⎠		%17
⎛
⎝		6 x4+x3−6 x2+5 x−6⎞
⎠		%17
0%170%171%17
⎛
⎝		−x3+3 x2+7⎞
⎠		%17
0%170%170%171%17
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
         
 
⎡
⎢
⎢
⎢
⎢
⎣
1%170%170%170%17
0%171%170%170%17
0%170%171%170%17
0%170%170%17
⎛
⎝		−x4−2 x3+8 x2−3 x+2⎞
⎠		%17
⎤
⎥
⎥
⎥
⎥
⎦
         

You can check this:

normal(U*A*V-D)
     
⎡
⎢
⎢
⎢
⎣
0000
0000
0000
0000
⎤
⎥
⎥
⎥
⎦
          
B:=[[x^2+x-1,1,0,1],[-1,x,0,-1],[0,x^2+1,x,0],[1,0,1,x^2+x+1]] % 3:; L:=smith(B)
     
 
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0%3
⎛
⎝		−1⎞
⎠		%3
0%30%3
1%30%30%3
⎛
⎝		−x2−x+1⎞
⎠		%3
0%3
⎛
⎝		x2+1⎞
⎠		%3
⎛
⎝		−x⎞
⎠		%3
⎛
⎝		x2+1⎞
⎠		%3
⎛
⎝		−1⎞
⎠		%3
⎛
⎝		−x4−x3+x+1⎞
⎠		%3
⎛
⎝		x3+x2−x+1⎞
⎠		%3
⎛
⎝		−x4−x3+x2−x⎞
⎠		%3
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
         
 
⎡
⎢
⎢
⎢
⎢
⎣
1%30%30%30%3
0%31%30%30%3
0%30%31%30%3
0%30%30%3
⎛
⎝		−x6+x5+x+1⎞
⎠		%3
⎤
⎥
⎥
⎥
⎥
⎦
,
         
 
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1%3x%3
⎛
⎝		x3+x2−x⎞
⎠		%3
⎛
⎝		−x7+x6+x4+x3+x2+x−1⎞
⎠		%3
0%31%3
⎛
⎝		x2+x−1⎞
⎠		%3
⎛
⎝		−x6+x5+x3+x2+x+1⎞
⎠		%3
0%30%31%3
⎛
⎝		−x4−x3−x2−x⎞
⎠		%3
0%30%30%31%3
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
         
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