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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 = | -11 -24 31 -5  |
     | 21  -43 7  -13 |
     | 41  -43 13 -17 |
     | -7  -48 2  8   |

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

       4      3      2
o4 = (x  + 33x  - 32x  + 9x - 41)

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

o5 = (| -33 1 0 0 |, | 0 -5  -6  22  |, | 46  14  -11 1 |)
      | 32  0 1 0 |  | 0 -41 36  -42 |  | -40 -49 21  0 |
      | -9  0 0 1 |  | 0 -33 -14 -8  |  | -49 5   41  0 |
      | 41  0 0 0 |  | 1 37  22  -7  |  | -27 4   -7  0 |

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

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

o7 = true

Ways to use rationalNormalForm :