This function produces a diagonal matrix
D, and invertible matrices
P and
Q such that
D = PMQ. Warning: even though this function is called the Smith normal form, it doesn't necessarily satisfy the more stringent condition that the diagonal entries
d1, d2, ..., dn of
D satisfy:
d1|d2|...|dn..
i1 : M = matrix{{1,2,3},{1,34,45},{2213,1123,6543},{0,0,0}}
o1 = | 1 2 3 |
| 1 34 45 |
| 2213 1123 6543 |
| 0 0 0 |
4 3
o1 : Matrix ZZ <--- ZZ
|
i2 : (D,P,Q) = smithNormalForm M
o2 = (| 135654 0 0 |, | 1 33471 -43292 0 |, | 171927 -42421 54868 |)
| 0 1 0 | | 0 1 0 0 | | 93042 -22957 29693 |
| 0 0 1 | | 0 0 1 0 | | -74119 18288 -23654 |
| 0 0 0 | | 0 0 0 1 |
o2 : Sequence
|
i3 : D == P * M * Q
o3 = true
|
i4 : (D,P) = smithNormalForm(M, ChangeMatrix=>{true,false})
o4 = (| 135654 0 0 |, | 1 33471 -43292 0 |)
| 0 1 0 | | 0 1 0 0 |
| 0 0 1 | | 0 0 1 0 |
| 0 0 0 | | 0 0 0 1 |
o4 : Sequence
|
i5 : D = smithNormalForm(M, ChangeMatrix=>{false,false}, KeepZeroes=>true)
o5 = | 135654 0 0 |
| 0 1 0 |
| 0 0 1 |
3 3
o5 : Matrix ZZ <--- ZZ
|
This function is the underlying routine used by minimalPresentation in the case when the ring is ZZ, or a polynomial ring in one variable over a field.
i6 : prune coker M
o6 = cokernel | 135654 |
| 0 |
2
o6 : ZZ-module, quotient of ZZ
|
In the following example, we test the result be checking that the entries of
D1, P1 M Q1 are the same. The degrees associated to these matrices do not match up, so a simple test of equality would return false.
i7 : S = ZZ/101[t]
o7 = S
o7 : PolynomialRing
|
i8 : D = diagonalMatrix{t^2+1, (t^2+1)^2, (t^2+1)^3, (t^2+1)^5}
o8 = | t2+1 0 0 0 |
| 0 t4+2t2+1 0 0 |
| 0 0 t6+3t4+3t2+1 0 |
| 0 0 0 t10+5t8+10t6+10t4+5t2+1 |
4 4
o8 : Matrix S <--- S
|
i9 : P = random(S^4, S^4)
o9 = | -3 -18 -6 30 |
| -17 -39 -6 -41 |
| -3 -28 34 9 |
| 14 32 20 -17 |
4 4
o9 : Matrix S <--- S
|
i10 : Q = random(S^4, S^4)
o10 = | 29 -33 37 -31 |
| 10 40 -6 -27 |
| 45 -2 37 34 |
| -36 38 28 -6 |
4 4
o10 : Matrix S <--- S
|
i11 : M = P*D*Q
o11 = | 31t10-47t8+40t6+27t4+9t2-1 29t10+44t8-t6+10t4-49t2+26
| -39t10+7t8+47t6+26t4+45t2+20 -43t10-13t8-14t6-35t4-11t2-20
| -21t10-4t8+7t6-41t4+31 39t10-7t8+19t6-25t4-29t2-40
| 6t10+30t8-50t6+50t4+39t2+16 -40t10+2t8-36t6-48t4-40t2+31
-----------------------------------------------------------------------
32t10-42t8-3t6-36t4+3t2+9 22t10+9t8+16t6-7t4-43t2-7 |
-37t10+17t8+14t6+6t4-2t2-48 44t10+18t8+34t6-28t4+19t2+6 |
50t10+48t8+41t6-2t4+7t2-49 47t10+33t8+10t6+48t4-45t2+32 |
29t10+44t8+20t6-5t4-26t2-16 t10+5t8-17t6-26t4-16t2-11 |
4 4
o11 : Matrix S <--- S
|
i12 : (D1,P1,Q1) = smithNormalForm M;
|
i13 : D1 - P1*M*Q1 == 0
o13 = true
|
i14 : prune coker M
o14 = cokernel | t10+5t8+10t6+10t4+5t2+1 0 0 0 |
| 0 t6+3t4+3t2+1 0 0 |
| 0 0 t4+2t2+1 0 |
| 0 0 0 t2+1 |
4
o14 : S-module, quotient of S
|
This routine is under development. The main idea is to compute a Gröbner basis, transpose the generators, and repeat, until we encounter a matrix whose transpose is already a Gröbner basis. This may depend heavily on the monomial order.