rationalNormalForm -- rational normal form of a matrix

• Usage:

  (B,P,Q) = rationalNormalForm A

  (B,P) = rationalNormalForm(A,ChangeMatrix=>{true,false})

  (B,Q) = rationalNormalForm(A,ChangeMatrix=>{false,true})

  B = rationalNormalForm(A,ChangeMatrix=>{false,false})

• Inputs:
  • A, a matrix, a square matrix with constant entries
• Optional inputs:
  • ChangeMatrix => ..., default value {true, true}
• Outputs:
  • B, a matrix, the rational normal form of A
  • P, a matrix, invertible (left) change of basis matrix
  • Q, a matrix, invertible (right) change of basis matrix

i1 : R = ZZ/101[x]

o1 = R

o1 : PolynomialRing

i2 : M = R^4

      4
o2 = R

o2 : R-module, free

i3 : A = random(M,M)

o3 = | 24  19  -8  -38 |
     | -36 19  -22 -16 |
     | -30 -10 -29 39  |
     | -29 -29 -24 21  |

             4      4
o3 : Matrix R  <-- R

i4 : factor det(x*id_M - A)

       2              2
o4 = (x  + 39x - 10)(x  + 27x - 27)

o4 : Expression of class Product

i5 : (B,P,Q) = rationalNormalForm A

o5 = (| -27 1 0   0 |, | 15 -11 46  -1 |, | 50  13  7   -23 |)
      | 27  0 0   0 |  | 12 -46 -47 39 |  | -37 -24 -41 -18 |
      | 0   0 -39 1 |  | -9 47  33  -2 |  | 23  1   33  1   |
      | 0   0 10  0 |  | 41 -40 22  50 |  | -8  0   -47 0   |

o5 : Sequence

i6 : B - P*A*Q == 0

o6 = true

i7 : P*Q - id_M == 0

o7 = true