rationalNormalForm -- rational normal form of a matrix

Synopsis

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 => ..., -- rational normal form of a matrix
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

Description

This function produces a matrix B in rational normal form, and invertible matrices P and Q such that P*Q = I and B = P*A*Q.

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 = | -33 -33 38  8   |
     | -37 1   -9  -29 |
     | -39 -30 -29 9   |
     | 21  -10 44  -16 |

             4       4
o3 : Matrix R  <--- R

i4 : factor det(x*id_M - A)

                       2
o4 = (x + 14)(x + 19)(x  + 44x - 3)

o4 : Expression of class Product

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

o5 = (| 1 0 0   0 |, | -22 -26 5   -42 |, | 30  -46 31  -19 |)
      | 0 1 0   0 |  | 43  -40 46  34  |  | 15  7   46  -42 |
      | 0 0 -44 1 |  | 9   15  -7  25  |  | -46 43  -48 1   |
      | 0 0 3   0 |  | 39  35  -10 -29 |  | 36  -16 -36 0   |

o5 : Sequence

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

o6 = true

i7 : P*Q - id_M == 0

o7 = false

Ways to use `rationalNormalForm` :

rationalNormalForm(Matrix)

rationalNormalForm -- rational normal form of a matrix

Synopsis

Description

Ways to use rationalNormalForm :

Ways to use `rationalNormalForm` :