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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 = | -7  -15 -15 50  |
     | 43  -46 -27 33  |
     | -1  -29 5   -29 |
     | -46 -40 -38 9   |

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

               3     2
o4 = (x + 41)(x  - 2x  - 49x + 20)

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

o5 = (| 1 0   0 0 |, | -7  47  -46 20  |, | 34  14 39  50 |)
      | 0 2   1 0 |  | -9  46  12  38  |  | -12 -9 -17 1  |
      | 0 49  0 1 |  | -4  -2  10  -24 |  | -13 47 22  0  |
      | 0 -20 0 0 |  | -38 -18 -17 -8  |  | 32  18 -17 0  |

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

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

o7 = false

Ways to use rationalNormalForm :