next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
Kronecker :: rationalNormalForm

rationalNormalForm -- rational normal form of a matrix

Synopsis

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 = | 25  -16 11  27  |
     | 45  -34 6   -48 |
     | -49 -2  -35 28  |
     | -21 -41 -6  -23 |

             4       4
o3 : Matrix R  <--- R
i4 : factor det(x*id_M - A)

       4      3      2
o4 = (x  - 34x  + 10x  - 2x - 45)

o4 : Expression of class Product
i5 : (B,P,Q) = rationalNormalForm A

o5 = (| 34  1 0 0 |, | 0 48  -5  -49 |, | 45  11 25  1 |)
      | -10 0 1 0 |  | 0 -33 18  46  |  | -22 6  45  0 |
      | 2   0 0 1 |  | 0 -17 -43 35  |  | 7   14 -49 0 |
      | 45  0 0 0 |  | 1 42  -9  16  |  | 21  23 -21 0 |

o5 : Sequence
i6 : B - P*A*Q == 0

o6 = true
i7 : P*Q - id_M == 0

o7 = true

Ways to use rationalNormalForm :