octave:1> size = 12
size =  12

octave:2> I = eye(size)
I =

Diagonal Matrix

   1   0   0   0   0   0   0   0   0   0   0   0
   0   1   0   0   0   0   0   0   0   0   0   0
   0   0   1   0   0   0   0   0   0   0   0   0
   0   0   0   1   0   0   0   0   0   0   0   0
   0   0   0   0   1   0   0   0   0   0   0   0
   0   0   0   0   0   1   0   0   0   0   0   0
   0   0   0   0   0   0   1   0   0   0   0   0
   0   0   0   0   0   0   0   1   0   0   0   0
   0   0   0   0   0   0   0   0   1   0   0   0
   0   0   0   0   0   0   0   0   0   1   0   0
   0   0   0   0   0   0   0   0   0   0   1   0
   0   0   0   0   0   0   0   0   0   0   0   1

octave:3> A = [I(12,:) ; I(1:11,:)] % downshift permutation
A =

   0   0   0   0   0   0   0   0   0   0   0   1
   1   0   0   0   0   0   0   0   0   0   0   0
   0   1   0   0   0   0   0   0   0   0   0   0
   0   0   1   0   0   0   0   0   0   0   0   0
   0   0   0   1   0   0   0   0   0   0   0   0
   0   0   0   0   1   0   0   0   0   0   0   0
   0   0   0   0   0   1   0   0   0   0   0   0
   0   0   0   0   0   0   1   0   0   0   0   0
   0   0   0   0   0   0   0   1   0   0   0   0
   0   0   0   0   0   0   0   0   1   0   0   0
   0   0   0   0   0   0   0   0   0   1   0   0
   0   0   0   0   0   0   0   0   0   0   1   0

octave:4> Z = zeros(size, size);

octave:5> G = [Z, A; A', Z];

octave:6> eig(G)
ans =

  -1
  -1
  -1
  -1
  -1
  -1
  -1
  -1
  -1
  -1
  -1
  -1
   1
   1
   1
   1
   1
   1
   1
   1
   1
   1
   1
   1

octave:7> eig([Z, A*A; (A*A)', Z])
ans =

  -1
  -1
  -1
  -1
  -1
  -1
  -1
  -1
  -1
  -1
  -1
  -1
   1
   1
   1
   1
   1
   1
   1
   1
   1
   1
   1
   1

octave:8> eig([Z, A*A*A; (A*A*A)', Z])
ans =

  -1
  -1
  -1
  -1
  -1
  -1
  -1
  -1
  -1
  -1
  -1
  -1
   1
   1
   1
   1
   1
   1
   1
   1
   1
   1
   1
   1

octave:9> eig([Z, eye(12); eye(12)', Z])
ans =

  -1
  -1
  -1
  -1
  -1
  -1
  -1
  -1
  -1
  -1
  -1
  -1
   1
   1
   1
   1
   1
   1
   1
   1
   1
   1
   1
   1

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