kroneckerIndices -- data which classify a matrix pencil up to strict equivalence

• Usage:

  (L, dualL, d) = kroneckerIndices(P,x,y)

• Inputs:
  • P, a matrix, a matrix of linear forms
  • x, a ring element
  • y, a ring element
• Outputs:
  • L, a list, degrees of minimal generators of the kernel
  • dualL, a list, degrees of minimal generators of the kernel of the dual
  • d, a list, elementary divisors

i1 : R = ZZ/101[x,y]

o1 = R

o1 : PolynomialRing

i2 : A = matrix{{x,y,x,y},{y,x,y,x},{x,y,x,y},{y,y,y,y},{x,x,y,y}}

o2 = | x y x y |
     | y x y x |
     | x y x y |
     | y y y y |
     | x x y y |

             5      4
o2 : Matrix R  <-- R

i3 : kroneckerNormalForm(A, ChangeMatrix => {false,false})

o3 = | 0 0 0   0   |
     | 0 x 0   0   |
     | 0 y 0   0   |
     | 0 0 x-y 0   |
     | 0 0 0   x-y |

             5      4
o3 : Matrix R  <-- R

i4 : kroneckerIndices(A,x,y)

o4 = ({0}, {0, 1}, {x - y, x - y})

o4 : Sequence