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 = | -9  1  1   -43 |
     | 19  23 50  -33 |
     | 3   1  17  33  |
     | -50 16 -47 43  |

             4       4
o3 : Matrix R  <--- R

i4 : factor det(x*id_M - A)

       2            2
o4 = (x  + 3x + 6)(x  + 24x + 32)

o4 : Expression of class Product

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

o5 = (| -3 1 0   0 |, | 32 -31 27 -45 |, | -39 -10 -48 -24 |)
      | -6 0 0   0 |  | 35 48  26 -17 |  | 1   -29 -37 38  |
      | 0  0 -24 1 |  | 8  -33 32 -25 |  | -42 1   -17 1   |
      | 0  0 -32 0 |  | 31 8   37 25  |  | -11 0   44  0   |

o5 : Sequence

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

o6 = true

i7 : P*Q - id_M == 0

o7 = true

Ways to use `rationalNormalForm` :

rationalNormalForm(Matrix)

rationalNormalForm -- rational normal form of a matrix

Synopsis

Description

Ways to use rationalNormalForm :

Ways to use `rationalNormalForm` :