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 = | -11 34  -11 -42 |
     | -4  -24 -37 39  |
     | -36 -4  36  -8  |
     | 24  46  45  -8  |

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

              3
o4 = (x + 7)(x  - 20x + 30)

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

o5 = (| -7 0   0 0 |, | 2  17 11  -14 |, | -22 -21 -24 42 |)
      | 0  0   1 0 |  | 41 -5 42  -49 |  | 26  -7  10  1  |
      | 0  20  0 1 |  | 12 1  -50 -43 |  | -44 32  -1  0  |
      | 0  -30 0 0 |  | 14 19 3   47  |  | 1   -8  44  0  |

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

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

o7 = true

Ways to use rationalNormalForm :