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Find power method for eigenvalue [[8,-6,2],[-6,7,-4],[2,-4,3]] [ Calculator, Method and examples ]

Solution:
Your problem -> power method for eigenvalue [[8,-6,2],[-6,7,-4],[2,-4,3]]

A=
 8 -6 2 -6 7 -4 2 -4 3

x_0=
 1 1 1

1^(st) Iteration

A x_0 =
 8 -6 2 -6 7 -4 2 -4 3

 1 1 1
=
 4 -3 1

and by scaling we obtain the approximation
x_1=1/4
 4 -3 1
=
 1 -0.75 0.25

2^(nd) Iteration

A x_1 =
 8 -6 2 -6 7 -4 2 -4 3

 1 -0.75 0.25
=
 13 -12.25 5.75

and by scaling we obtain the approximation
x_2=1/13
 13 -12.25 5.75
=
 1 -0.9423 0.4423

3^(rd) Iteration

A x_2 =
 8 -6 2 -6 7 -4 2 -4 3

 1 -0.9423 0.4423
=
 14.5385 -14.3654 7.0962

and by scaling we obtain the approximation
x_3=1/14.5385
 14.5385 -14.3654 7.0962
=
 1 -0.9881 0.4881

4^(th) Iteration

A x_3 =
 8 -6 2 -6 7 -4 2 -4 3

 1 -0.9881 0.4881
=
 14.9048 -14.869 7.4167

and by scaling we obtain the approximation
x_4=1/14.9048
 14.9048 -14.869 7.4167
=
 1 -0.9976 0.4976

5^(th) Iteration

A x_4 =
 8 -6 2 -6 7 -4 2 -4 3

 1 -0.9976 0.4976
=
 14.9808 -14.9736 7.4832

and by scaling we obtain the approximation
x_5=1/14.9808
 14.9808 -14.9736 7.4832
=
 1 -0.9995 0.4995

6^(th) Iteration

A x_5 =
 8 -6 2 -6 7 -4 2 -4 3

 1 -0.9995 0.4995
=
 14.9962 -14.9947 7.4966

and by scaling we obtain the approximation
x_6=1/14.9962
 14.9962 -14.9947 7.4966
=
 1 -0.9999 0.4999

7^(th) Iteration

A x_6 =
 8 -6 2 -6 7 -4 2 -4 3

 1 -0.9999 0.4999
=
 14.9992 -14.9989 7.4993

and by scaling we obtain the approximation
x_7=1/14.9992
 14.9992 -14.9989 7.4993
=
 1 -1 0.5

:. The dominant eigenvalue lamda=14.9992~=15

and the dominant eigenvector is :
=
 1 -1 0.5
~=
 1 -1 0.5

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