Compare matrices consistently for different base rings

[SimonKing]

Apparently we use three essentially different ways of comparing matrices.

  sage: def test(K):
  ....:     M1 = Matrix(K,[[1,0,0]])
  ....:     M2 = Matrix(K,[[0,1,0]])
  ....:     M3 = Matrix(K,[[0,1,1]])
  ....:     M4 = Matrix(K,[[0,0,1]])
  ....:     L = [M1,M2,M3,M4]
  ....:     L.sort()
  ....:     return L
  ....:

First group

Matrices over not too big fields of characteristic 2

  sage: K = GF(2)
  sage: test(K)
  [[1 0 0], [0 1 0], [0 0 1], [0 1 1]]
  sage: K = GF(4,'z')
  sage: test(K)
  [[1 0 0], [0 1 0], [0 0 1], [0 1 1]]
  sage: K = GF(2^10,'q')
  sage: test(K)
  [[1 0 0], [0 1 0], [0 0 1], [0 1 1]]

Second group

Matrices over big finite fields of characteristic 2

  sage: K = GF(2^100,'q')
  sage: test(K)
  [[1 0 0], [0 1 0], [0 1 1], [0 0 1]]

Third group

Matrices over all other rings

  sage: K = GF(3)
  sage: test(K)
  [[0 0 1], [0 1 0], [0 1 1], [1 0 0]]
  sage: K = GF(9,'z')
  sage: test(K)
  [[0 0 1], [0 1 0], [0 1 1], [1 0 0]]
  sage: K = GF(3^100,'q')
  sage: test(K)
  [[0 0 1], [0 1 0], [0 1 1], [1 0 0]]
  sage: K = QQ
  sage: test(K)
  [[0 0 1], [0 1 0], [0 1 1], [1 0 0]]
  sage: K = ZZ
  sage: test(K)
  [[0 0 1], [0 1 0], [0 1 1], [1 0 0]]
  sage: K.<x> = GF(2)[]
  sage: test(K)
  [[0 0 1], [0 1 0], [0 1 1], [1 0 0]]
  sage: K.<x> = GF(3)[]
  sage: test(K)
  [[0 0 1], [0 1 0], [0 1 1], [1 0 0]]
  sage: K.<x> = GF(4,'z')[]
  sage: test(K)
  [[0 0 1], [0 1 0], [0 1 1], [1 0 0]]
  sage: K.<x> = GF(2^100,'z')[]
  sage: test(K)
  [[0 0 1], [0 1 0], [0 1 1], [1 0 0]]

I guess the inconsistent behaviour is "big fun" for all who want to implement the F4 algorithm in Sage. It ought to be fixed.

[malb]

With #13180 fixed this boils down to two groups: M4RI(E) vs the rest.