# LA+Chapter+8+EX+6+solved+by+Leenaeul

## 2419 days ago by math2013

A=matrix(QQ,3,3,[1,2,2,2,1,-2,2,-2,1]) print A.eigenvalues() print print A.eigenvectors_right() x1=vector([1,-1,-1]) x2=vector([1,0,1]) x3=vector([0,1,-1]) print print x1.norm() print x2.norm() print x3.norm() x11 = vector([1/sqrt(3), -1/sqrt(3), -1/sqrt(3)]) x22 = vector([1/sqrt(2), 0/sqrt(2), 1/sqrt(2)]) x33 = vector([0/sqrt(2), 1/sqrt(2), -1/sqrt(2)]) print print x11 print x22 print x33 print P = matrix([x11, x22, x33]) print P D = P.inverse()*A*P print print D
 [-3, 3, 3] [(-3, [ (1, -1, -1) ], 1), (3, [ (1, 0, 1), (0, 1, -1) ], 2)] sqrt(3) sqrt(2) sqrt(2) (1/3*sqrt(3), -1/3*sqrt(3), -1/3*sqrt(3)) (1/2*sqrt(2), 0, 1/2*sqrt(2)) (0, 1/2*sqrt(2), -1/2*sqrt(2)) [ 1/3*sqrt(3) -1/3*sqrt(3) -1/3*sqrt(3)] [ 1/2*sqrt(2) 0 1/2*sqrt(2)] [ 0 1/2*sqrt(2) -1/2*sqrt(2)] [ 1/9*(6*sqrt(2) + sqrt(3))*sqrt(3) + 1/3*sqrt(2)*sqrt(3) -1/6*(3*sqrt(2) - 2*sqrt(3))*sqrt(2) - 1/9*(6*sqrt(2) + sqrt(3))*sqrt(3) 1/6*(3*sqrt(2) - 2*sqrt(3))*sqrt(2) - 1/9*(6*sqrt(2) + sqrt(3))*sqrt(3) + 1/3*sqrt(2)*sqrt(3)] [ -1/6*(3*sqrt(2) + 2*sqrt(3))*sqrt(2) + 1/9*(6*sqrt(2) - sqrt(3))*sqrt(3) -1/9*(6*sqrt(2) - sqrt(3))*sqrt(3) - 1/3*sqrt(2)*sqrt(3) -1/6*(3*sqrt(2) + 2*sqrt(3))*sqrt(2) - 1/9*(6*sqrt(2) - sqrt(3))*sqrt(3) + 1/3*sqrt(2)*sqrt(3)] [ 1/6*(3*sqrt(2) - 2*sqrt(3))*sqrt(2) - 1/3 -1/6*(3*sqrt(2) + 2*sqrt(3))*sqrt(2) + 1/3 1/6*(3*sqrt(2) - 2*sqrt(3))*sqrt(2) + 1/6*(3*sqrt(2) + 2*sqrt(3))*sqrt(2) + 1/3] [-3, 3, 3] [(-3, [ (1, -1, -1) ], 1), (3, [ (1, 0, 1), (0, 1, -1) ], 2)] sqrt(3) sqrt(2) sqrt(2) (1/3*sqrt(3), -1/3*sqrt(3), -1/3*sqrt(3)) (1/2*sqrt(2), 0, 1/2*sqrt(2)) (0, 1/2*sqrt(2), -1/2*sqrt(2)) [ 1/3*sqrt(3) -1/3*sqrt(3) -1/3*sqrt(3)] [ 1/2*sqrt(2) 0 1/2*sqrt(2)] [ 0 1/2*sqrt(2) -1/2*sqrt(2)] [ 1/9*(6*sqrt(2) + sqrt(3))*sqrt(3) + 1/3*sqrt(2)*sqrt(3) -1/6*(3*sqrt(2) - 2*sqrt(3))*sqrt(2) - 1/9*(6*sqrt(2) + sqrt(3))*sqrt(3) 1/6*(3*sqrt(2) - 2*sqrt(3))*sqrt(2) - 1/9*(6*sqrt(2) + sqrt(3))*sqrt(3) + 1/3*sqrt(2)*sqrt(3)] [ -1/6*(3*sqrt(2) + 2*sqrt(3))*sqrt(2) + 1/9*(6*sqrt(2) - sqrt(3))*sqrt(3) -1/9*(6*sqrt(2) - sqrt(3))*sqrt(3) - 1/3*sqrt(2)*sqrt(3) -1/6*(3*sqrt(2) + 2*sqrt(3))*sqrt(2) - 1/9*(6*sqrt(2) - sqrt(3))*sqrt(3) + 1/3*sqrt(2)*sqrt(3)] [ 1/6*(3*sqrt(2) - 2*sqrt(3))*sqrt(2) - 1/3 -1/6*(3*sqrt(2) + 2*sqrt(3))*sqrt(2) + 1/3 1/6*(3*sqrt(2) - 2*sqrt(3))*sqrt(2) + 1/6*(3*sqrt(2) + 2*sqrt(3))*sqrt(2) + 1/3]