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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 = | -38 -30 50  -30 |
     | 12  0   20  -41 |
     | 42  -36 -21 2   |
     | 33  19  -1  -28 |

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

               3      2
o4 = (x + 12)(x  - 26x  - 22x - 39)

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

o5 = (| 1 0  0 0 |, | -29 48  23  10  |, | -21 -33 29 33 |)
      | 0 26 1 0 |  | -45 -30 -32 -9  |  | 35  -1  -8 1  |
      | 0 22 0 1 |  | -30 -20 -24 -28 |  | 7   16  37 0  |
      | 0 39 0 0 |  | -42 -27 43  28  |  | 42  44  -3 0  |

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

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

o7 = false

Ways to use rationalNormalForm :