>>>% octave
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octave:1> A=[2 -1 0 1; 0 3 -1 0; 0 1 1 0; 0 -1 0 3]
A =

   2  -1   0   1
   0   3  -1   0
   0   1   1   0
   0  -1   0   3

octave:2> eig(A)
ans =

   2.0000
   3.0000
   2.0000
   2.0000

octave:3> B=A-2*eye(4)
B =

   0  -1   0   1
   0   1  -1   0
   0   1  -1   0
   0  -1   0   1

octave:4> rref(B)
ans =

   0   1   0  -1
   0   0   1  -1
   0   0   0   0
   0   0   0   0

octave:5> rref(B^2)
ans =

        0   1.0000  -0.5000  -0.5000
        0        0        0        0
        0        0        0        0
        0        0        0        0

octave:6> rref(B^3)
ans =

        0   1.0000  -0.5000  -0.5000
        0        0        0        0
        0        0        0        0
        0        0        0        0

octave:7> v1=[0 1 0 2]'
v1 =

   0
   1
   0
   2

octave:8> v2=B*v1
v2 =

   1
   1
   1
   1

octave:9> B*v2
ans =

   0
   0
   0
   0

octave:10>