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A = matrix([[76,3,-31,68,56],[-84,32,58,-129,-103],[4,10,-19,58,-15],[25,13,-105,181,41],[-27,4,65,-89,5]]); A
[ 76 3 -31 68 56] [ -84 32 58 -129 -103] [ 4 10 -19 58 -15] [ 25 13 -105 181 41] [ -27 4 65 -89 5] 
A.eigenvalues()
[55, 55, 55, 55, 55] 
A.eigenmatrix_right()[1]
[ 1 0 0 0 0] [-4 0 0 0 0] [ 7 0 0 0 0] [ 8 0 0 0 0] [-6 0 0 0 0] 
S = column_matrix([[1,-4,7,8,-6],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]); S
[ 1 0 0 0 0] [-4 1 0 0 0] [ 7 0 1 0 0] [ 8 0 0 1 0] [-6 0 0 0 1] 
A1 = S.inverse()*A*S; A1
[ 55 3 -31 68 56] [ 0 44 -66 143 121] [ 0 -11 198 -418 -407] [ 0 -11 143 -363 -407] [ 0 22 -121 319 341] 
A1[1:,1:].eigenvalues()
[55, 55, 55, 55] 
A1[1:,1:].eigenmatrix_right()[1]
[ 1 0 0 0] [-3 0 0 0] [-3 0 0 0] [ 2 0 0 0] 
S1 = column_matrix([[1,-4,7,8,-6],[0,1,-3,-3,2],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]); S1
[ 1 0 0 0 0] [-4 1 0 0 0] [ 7 -3 1 0 0] [ 8 -3 0 1 0] [-6 2 0 0 1] 
A2 = S1.inverse()*A*S1; A2
[ 55 4 -31 68 56] [ 0 55 -66 143 121] [ 0 0 0 11 -44] [ 0 0 -55 66 -44] [ 0 0 11 33 99] 
A2[2:,2:].eigenvalues()
[55, 55, 55] 
A2[2:,2:].eigenmatrix_right()[1]
[ 1 0 0] [ 1 0 0] [-1 0 0] 
S2 = column_matrix([[1,-4,7,8,-6],[0,1,-3,-3,2],[0,0,1,1,-1],[0,0,0,1,0],[0,0,0,0,1]]); S2
[ 1 0 0 0 0] [-4 1 0 0 0] [ 7 -3 1 0 0] [ 8 -3 1 1 0] [-6 2 -1 0 1] 
A3 = S2.inverse()*A*S2; A3
[ 55 4 -19 68 56] [ 0 55 -44 143 121] [ 0 0 55 11 -44] [ 0 0 0 55 0] [ 0 0 0 44 55] 
A3[3:,3:].eigenvalues()
[55, 55] 
A3[3:,3:].eigenmatrix_right()[1]
[0 0] [1 0] 
S3 = column_matrix([[1,-4,7,8,-6],[0,1,-3,-3,2],[0,0,1,1,-1],[0,0,0,0,1],[0,0,0,1,0]]); S3
[ 1 0 0 0 0] [-4 1 0 0 0] [ 7 -3 1 0 0] [ 8 -3 1 0 1] [-6 2 -1 1 0] 
A4 = S3.inverse()*A*S3; A4
[ 55 4 -19 56 68] [ 0 55 -44 121 143] [ 0 0 55 -44 11] [ 0 0 0 55 44] [ 0 0 0 0 55]