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 = | -33 1 -21 22 |
| 27 -8 -49 -23 |
| 20 -7 -7 -48 |
| -34 15 1 -7 |
4 4
o9 : Matrix S <--- S
|
i10 : Q = random(S^4, S^4)
o10 = | 16 35 0 -10 |
| -30 35 27 -5 |
| -21 35 -11 15 |
| -11 -38 44 44 |
4 4
o10 : Matrix S <--- S
|
i11 : M = P*D*Q
o11 = | -40t10+2t8+41t6-16t4+30t2+45 -28t10-39t8-5t6-26t4+4t2+36
| -50t10-48t8+24t6-t4+12t2+35 -35t10+27t8-45t6-18t4+14t2+26
| 23t10+14t8-27t6-28t4-17t2-7 6t10+30t8+17t6-11t4+10t2+14
| -24t10-19t8+42t6-46t4-11t2-29 -37t10+17t8-32t6-43t4-18t2+40
-----------------------------------------------------------------------
-42t10-8t8+13t6-3t4+32t2+14 -42t10-8t8-28t6+44t4-27t2-32 |
-2t10-10t8+14t6-33t4-37t2+18 -2t10-10t8-48t6+37t4+19t2+43 |
9t10+45t8-35t6+31t4-t2-2 9t10+45t8-15t6+12t4+4t2+42 |
-5t10-25t8+40t6+19t4+45t2-15 -5t10-25t8-35t6+21t4+8t2-28 |
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.