box% octave
GNU Octave, version 5.2.0
Copyright (C) 2020 John W. Eaton and others.
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octave:1> M = [1 0 0 0 1; 0 1 1 1 0; 0 1 1 1 0; 0 1 1 1 0; 1 0 0 0 1]
M =

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

octave:2> I = eye(5)
I =

Diagonal Matrix

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

octave:3> P = [I(:,2) I(:,3) I(:,4) I(:,1) I(:,5)]
P =

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

octave:4> P'*M*P
ans =

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

octave:5> eigs(M)
ans =

   3.00000
   2.00000
  -0.00000
   0.00000
   0.00000

octave:6> eigs([1 1 1; 1 1 1; 1 1 1])
ans =

   3.0000e+00
  -5.6242e-16
   7.3053e-18

octave:7> eigs([1 1; 1 1])
ans =

   2
   0

octave:8>