-- -*- M2-comint -*- {* hash: 1486722414 *}

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

i6 : prune coker M

o6 = cokernel | 135654 |
              | 0      |

                              2
o6 : ZZ-module, quotient of ZZ

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 = | 5   23  -6 -15 |
     | 48  -12 15 -8  |
     | -27 1   12 49  |
     | -8  -21 32 3   |

             4       4
o9 : Matrix S  <--- S

i10 : Q = random(S^4, S^4)

o10 = | 16  -2 -23 -48 |
      | 7   32 -50 49  |
      | -32 32 33  -25 |
      | 50  43 49  -50 |

              4       4
o10 : Matrix S  <--- S

i11 : M = P*D*Q

o11 = | -43t10-13t8-36t6+4t4-45t2-14 -39t10+7t8+24t6-28t4-16t2-10 
      | 4t10+20t8-36t6+31t4-12t2+6   -41t10-3t8-31t6+40t4-33t2-41 
      | 26t10+29t8-23t6+24t4-26t2+25 -14t10+31t8+42t6+34t4-12t2-49
      | 49t10+43t8-29t6-2t4-17t2-38  28t10+39t8-9t6-47t4-35t2-8   
      -----------------------------------------------------------------------
      -28t10-39t8+27t6-4t4-18t2+24 43t10+13t8-26t6-13t4-48t2-31  |
      12t10-41t8+9t6-17t4+25t2+3   -4t10-20t8-11t6-36t4+21t2-39  |
      -23t10-14t8-36t6-t4-22t2+35  -26t10-29t8+46t6-40t2+9       |
      46t10+28t8+t6+32t4+26t2+13   -49t10-43t8+23t6+20t4+24t2+21 |

              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

i15 :