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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 = | 5   16  1   12 |
     | -24 9   40  8  |
     | -43 -33 13  20 |
     | -48 -3  -31 24 |

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

       4      3     2
o4 = (x  + 50x  + 4x  - 38x + 27)

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

o5 = (| -50 1 0 0 |, | 0 -25 -49 8   |, | 27 32  5   1 |)
      | -4  0 1 0 |  | 0 17  8   18  |  | -7 -16 -24 0 |
      | 38  0 0 1 |  | 0 38  30  -48 |  | -7 -33 -43 0 |
      | -27 0 0 0 |  | 1 42  8   -47 |  | 49 13  -48 0 |

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

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

o7 = true

Ways to use rationalNormalForm :