exercise3:
[[0.5 1.  0.  0.  0.  0.  0. ]
 [0.  0.5 1.  0.  0.  0.  0. ]
 [0.  0.  0.5 1.  0.  0.  0. ]
 [0.  0.  0.  0.5 1.  0.  0. ]
 [0.  0.  0.  0.  0.5 1.  0. ]
 [0.  0.  0.  0.  0.  0.5 1. ]
 [0.  0.  0.  0.  0.  0.  0.5]]

eigenvalues:
[0.5 0.5 0.5 0.5 0.5 0.5 0.5]
spectral radius rho =  0.5

eigenvectors:
[[ 1.00000000e+00 -1.00000000e+00  1.00000000e+00 -1.00000000e+00
   1.00000000e+00 -1.00000000e+00  1.00000000e+00]
 [ 0.00000000e+00  1.11022302e-16 -1.11022302e-16  1.11022302e-16
  -1.11022302e-16  1.11022302e-16 -1.11022302e-16]
 [ 0.00000000e+00  0.00000000e+00  1.23259516e-32 -1.23259516e-32
   1.23259516e-32 -1.23259516e-32  1.23259516e-32]
 [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  1.36845553e-48
  -1.36845553e-48  1.36845553e-48 -1.36845553e-48]
 [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
   1.51929084e-64 -1.51929084e-64  1.51929084e-64]
 [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
   0.00000000e+00  1.68675167e-80 -1.68675167e-80]
 [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
   0.00000000e+00  0.00000000e+00  1.87267054e-96]]
||Ax||2 / ||A||2 / ||x||2 =  0.9537138140361154
||Ax||2 / rho / ||x||2 =  2.8030595529069404