LA Chapter 8 Problem 4 solved by 김준현 revised by 이반석 finalized by Lee Naeul

A = matrix([[3,-4],[-4,3]]) print A print print A.eigenvalues() E = identity_matrix(2) print print (A-7*E).echelon_form() print print (A-(-1)*E).echelon_form() x1 = vector([1,-1]) x2 = vector([1,1]) P = column_matrix([x1, x2]) print print P print print P^-1*A*P 

[ 3 -4]
[-4  3]

[7, -1]

[4 4]
[0 0]

[ 4 -4]
[ 0  0]

[ 1  1]
[-1  1]

[ 7  0]
[ 0 -1]

[ 3 -4]
[-4  3]

[7, -1]

[4 4]
[0 0]

[ 4 -4]
[ 0  0]

[ 1  1]
[-1  1]

[ 7  0]
[ 0 -1]

B = matrix([[2,1,1],[1,2,1], [1,1,2]]) print B print print B.eigenvalues() E = identity_matrix(3) print print (B-4*E).echelon_form() print print (B-E).echelon_form() print print (B-E).echelon_form() x1 = vector([1,1,1]) x2 = vector([-1,1,0]) x3 = vector([-1,0,1]) P = column_matrix([x1, x2, x3]) print print P print print P^-1*B*P 

[2 1 1]
[1 2 1]
[1 1 2]

[4, 1, 1]

[ 1  1 -2]
[ 0  3 -3]
[ 0  0  0]

[1 1 1]
[0 0 0]
[0 0 0]

[1 1 1]
[0 0 0]
[0 0 0]

[ 1 -1 -1]
[ 1  1  0]
[ 1  0  1]

[4 0 0]
[0 1 0]
[0 0 1]

[2 1 1]
[1 2 1]
[1 1 2]

[4, 1, 1]

[ 1  1 -2]
[ 0  3 -3]
[ 0  0  0]

[1 1 1]
[0 0 0]
[0 0 0]

[1 1 1]
[0 0 0]
[0 0 0]

[ 1 -1 -1]
[ 1  1  0]
[ 1  0  1]

[4 0 0]
[0 1 0]
[0 0 1]