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 = | -21 41 17 31 |
| -14 -31 41 -8 |
| -33 -29 43 -30 |
| 32 -3 32 -13 |
4 4
o9 : Matrix S <--- S
|
i10 : Q = random(S^4, S^4)
o10 = | 1 -15 41 17 |
| 18 14 -36 18 |
| 20 -48 8 39 |
| 21 -10 -19 -30 |
4 4
o10 : Matrix S <--- S
|
i11 : M = P*D*Q
o11 = | 45t10+23t8-18t6-14t4-27t2-9 -7t10-35t8+23t6-25t4-10t2-35
| 34t10-32t8+49t6+20t4-15t2-21 -21t10-4t8+44t6+17t4-t2+9
| -24t10-19t8+14t6-31t2-22 -3t10-15t8+27t6+38t4+41t2+42
| 30t10+49t8+31t6+45t4-26t2+42 29t10+44t8-34t6-17t4+23t2-9
-----------------------------------------------------------------------
17t10-16t8+3t6+11t4+13t2+38 -21t10-4t8+49t6-8t4-27t2+13 |
-50t10-48t8+30t6-16t4-32t2+12 38t10-12t8-41t6-27t4-3t2+33 |
-36t10+22t8-16t6-t4-29t2-1 -9t10-45t8-29t6-25t4+48t2-21 |
45t10+23t8-t6+13t4-4t2+4 -14t10+31t8-3t6+15t4-31t2+7 |
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.