Description
This function produces a diagonal matrix
D, and invertible matrices
P and
Q such that
D = PMQwith diagonal entries
d1, d2, ..., dn of
D satisfying
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 = (| 1 0 0 |, | 1 -102182 -43292 0 |, | 129506 17567889865
| 0 1 0 | | -1 102183 -92361 0 | | 70085 9507247241
| 0 0 135654 | | 1 -102183 92362 0 | | -55831 -7573648009
| 0 0 0 | | 0 0 0 1 |
------------------------------------------------------------------------
2383136963911713 |)
1289686610013066 |
-1027388073388531 |
o2 : Sequence
|
i3 : D == P * M * Q
o3 = true
|
i4 : (D,P) = smithNormalForm(M, ChangeMatrix=>{true,false}, KeepZeroes=>false)
o4 = (| 1 0 0 |, | 1 -102182 -43292 0 |)
| 0 1 0 | | -1 102183 -92361 0 |
| 0 0 135654 | | 1 -102183 92362 0 |
o4 : Sequence
|
i5 : D = smithNormalForm(M, ChangeMatrix=>{false,false})
o5 = | 1 0 0 |
| 0 1 0 |
| 0 0 135654 |
| 0 0 0 |
4 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 = | 24 19 -8 -38 |
| -36 19 -22 -16 |
| -30 -10 -29 39 |
| -29 -29 -24 21 |
4 4
o9 : Matrix S <-- S
|
i10 : Q = random(S^4, S^4)
o10 = | 34 -18 -28 16 |
| 19 -13 -47 22 |
| -47 -43 38 45 |
| -39 -15 2 -34 |
4 4
o10 : Matrix S <-- S
|
i11 : M = P*D*Q
o11 = | -33t10+37t8+46t6+48t4-24t2+5 -36t10+22t8-16t6+21t4+27t2+33
| 18t10-11t8+2t6+7t4-37t2-13 38t10-12t8+13t6+42t4-50t2-29
| -6t10-30t8-10t6+t4+33t2+46 21t10+4t8+43t6+41t4+19
| -11t10+46t8+8t6-4t4+29t2-16 -12t10+41t8+3t6+20t4-31t2
-----------------------------------------------------------------------
25t10+24t8+47t6-40t4-13t2-26 -21t10-4t8+36t6+37t4+35t2+17 |
-32t10+42t8-45t6+16t4-12t2-46 39t10-7t8+6t6-41t4+10t2+2 |
-23t10-14t8-19t6-36t4-25t2-17 -13t10+36t8-21t6-23t4+49t2+2 |
42t10+8t8+13t6-44t4+2t2-8 -7t10-35t8-39t6-9t4+35t2+33 |
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 | 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
o14 : S-module, quotient of S
|
The key idea of the current implementation is as follows: compute a Gröbner basis, transpose the generators, and repeat, until we encounter a matrix whose transpose is already a Gröbner basis.