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 = | -48 -7  22  19  |
     | 7   12  -47 -27 |
     | 0   -43 -42 -21 |
     | -15 7   16  48  |

             4       4
o3 : Matrix R  <--- R

i4 : factor det(x*id_M - A)

               3      2
o4 = (x - 32)(x  - 39x  + 34x - 19)

o4 : Expression of class Product

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

o5 = (| 1 0   0 0 |, | -17 -2  36  -18 |, | -44 39  -37 -12 |)
      | 0 39  1 0 |  | -23 27  -20 -29 |  | 40  -10 29  1   |
      | 0 -34 0 1 |  | -1  -12 -13 3   |  | -43 -35 -43 0   |
      | 0 19  0 0 |  | 46  48  -49 39  |  | -41 -44 -15 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` :