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