Power method for eigen values and eigen vectors, largest eigen value

1. Numerical Methods
Power Method for Eigen values
Dr. N. B. Vyas
Department of Mathematics,
Atmiya Institute of Technology & Science,
Rajkot (Gujarat) - INDIA
niravbvyas@gmail.com
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

2. Eigen values and Eigen vectors by iteration
Power Method
Power method is particularly useful for estimating numerically
largest or smallest eigenvalue and its corresponding eigenvector.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

3. Eigen values and Eigen vectors by iteration
Power Method
Power method is particularly useful for estimating numerically
largest or smallest eigenvalue and its corresponding eigenvector.
The intermediate (remaining) eigenvalues can also be found.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

4. Eigen values and Eigen vectors by iteration
Power Method
Power method is particularly useful for estimating numerically
largest or smallest eigenvalue and its corresponding eigenvector.
The intermediate (remaining) eigenvalues can also be found.
The power method, which is an iterative method, can be used
when
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

5. Eigen values and Eigen vectors by iteration
Power Method
Power method is particularly useful for estimating numerically
largest or smallest eigenvalue and its corresponding eigenvector.
The intermediate (remaining) eigenvalues can also be found.
The power method, which is an iterative method, can be used
when
(i) The matrix A of order n has n linearly independent eigenvectors.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

6. Eigen values and Eigen vectors by iteration
Power Method
Power method is particularly useful for estimating numerically
largest or smallest eigenvalue and its corresponding eigenvector.
The intermediate (remaining) eigenvalues can also be found.
The power method, which is an iterative method, can be used
when
(i) The matrix A of order n has n linearly independent eigenvectors.
(ii) The eigenvalues can be ordered in magnitude as
|λ1| > |λ2| ≥ |λ3| ≥ . . . ≥ |λn|
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

7. Eigen values and Eigen vectors by iteration
Power Method
Power method is particularly useful for estimating numerically
largest or smallest eigenvalue and its corresponding eigenvector.
The intermediate (remaining) eigenvalues can also be found.
The power method, which is an iterative method, can be used
when
(i) The matrix A of order n has n linearly independent eigenvectors.
(ii) The eigenvalues can be ordered in magnitude as
|λ1| > |λ2| ≥ |λ3| ≥ . . . ≥ |λn|
When this ordering is adopted, the eigenvalue λ1 with the
greatest magnitude is called the dominant eigenvalue of the
matrix A
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

8. Eigen values and Eigen vectors by iteration
Power Method
Power method is particularly useful for estimating numerically
largest or smallest eigenvalue and its corresponding eigenvector.
The intermediate (remaining) eigenvalues can also be found.
The power method, which is an iterative method, can be used
when
(i) The matrix A of order n has n linearly independent eigenvectors.
(ii) The eigenvalues can be ordered in magnitude as
|λ1| > |λ2| ≥ |λ3| ≥ . . . ≥ |λn|
When this ordering is adopted, the eigenvalue λ1 with the
greatest magnitude is called the dominant eigenvalue of the
matrix A
And the remaining eigenvalues λ2, λ3, . . . , λn are called the
subdominant eigenvalues of A.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

9. Eigen values and Eigen vectors by iteration
Power Method: Working rules for determining largest eigenvalue.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

10. Eigen values and Eigen vectors by iteration
Power Method: Working rules for determining largest eigenvalue.
Let A = [aij] be a matrix of order n × n.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

11. Eigen values and Eigen vectors by iteration
Power Method: Working rules for determining largest eigenvalue.
Let A = [aij] be a matrix of order n × n.
We start from any vector x0(= 0) with n components such that
Ax0 = x
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

12. Eigen values and Eigen vectors by iteration
Power Method: Working rules for determining largest eigenvalue.
Let A = [aij] be a matrix of order n × n.
We start from any vector x0(= 0) with n components such that
Ax0 = x
In order to get a convergent sequence of eigenvectors
simultaneously scaling method is adopted.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

13. Eigen values and Eigen vectors by iteration
Power Method: Working rules for determining largest eigenvalue.
Let A = [aij] be a matrix of order n × n.
We start from any vector x0(= 0) with n components such that
Ax0 = x
In order to get a convergent sequence of eigenvectors
simultaneously scaling method is adopted.
In which at each stage each components of the resultant
approximate vector is to be divided by its absolutely largest
component.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

14. Eigen values and Eigen vectors by iteration
Power Method: Working rules for determining largest eigenvalue.
Let A = [aij] be a matrix of order n × n.
We start from any vector x0(= 0) with n components such that
Ax0 = x
In order to get a convergent sequence of eigenvectors
simultaneously scaling method is adopted.
In which at each stage each components of the resultant
approximate vector is to be divided by its absolutely largest
component.
Then use the scaled vector in the next step.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

15. Eigen values and Eigen vectors by iteration
Power Method: Working rules for determining largest eigenvalue.
Let A = [aij] be a matrix of order n × n.
We start from any vector x0(= 0) with n components such that
Ax0 = x
In order to get a convergent sequence of eigenvectors
simultaneously scaling method is adopted.
In which at each stage each components of the resultant
approximate vector is to be divided by its absolutely largest
component.
Then use the scaled vector in the next step.
This absolutely largest component is known as numerically
largest eigenvalue.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

16. Eigen values and Eigen vectors by iteration
Accordingly x in eq -(1) can be scaled by dividing each of its
components by absolutely largest component of it. Thus
Ax0 = x = λ1x1; x1 is the scaled vector of x
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

17. Eigen values and Eigen vectors by iteration
Accordingly x in eq -(1) can be scaled by dividing each of its
components by absolutely largest component of it. Thus
Ax0 = x = λ1x1; x1 is the scaled vector of x
Now scaled vector x1 is to be used in the next iteration to obtain
Ax1 = x = λ2x2
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

18. Eigen values and Eigen vectors by iteration
Accordingly x in eq -(1) can be scaled by dividing each of its
components by absolutely largest component of it. Thus
Ax0 = x = λ1x1; x1 is the scaled vector of x
Now scaled vector x1 is to be used in the next iteration to obtain
Ax1 = x = λ2x2
Proceeding in this way, finally we get Axn = λn+1xn+1; where
n = 0, 1, 2, 3, ... Where λn+1 is the numerically largest eigenvalue
upto desired accuracy and xn+1 is the corresponding eigenvector.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

19. Eigen values and Eigen vectors by iteration
Accordingly x in eq -(1) can be scaled by dividing each of its
components by absolutely largest component of it. Thus
Ax0 = x = λ1x1; x1 is the scaled vector of x
Now scaled vector x1 is to be used in the next iteration to obtain
Ax1 = x = λ2x2
Proceeding in this way, finally we get Axn = λn+1xn+1; where
n = 0, 1, 2, 3, ... Where λn+1 is the numerically largest eigenvalue
upto desired accuracy and xn+1 is the corresponding eigenvector.
NOTE : The initial vector x0 is usually taken as a vector with
all components equal to 1.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

20. Eigen values and Eigen vectors by iteration
Accordingly x in eq -(1) can be scaled by dividing each of its
components by absolutely largest component of it. Thus
Ax0 = x = λ1x1; x1 is the scaled vector of x
Now scaled vector x1 is to be used in the next iteration to obtain
Ax1 = x = λ2x2
Proceeding in this way, finally we get Axn = λn+1xn+1; where
n = 0, 1, 2, 3, ... Where λn+1 is the numerically largest eigenvalue
upto desired accuracy and xn+1 is the corresponding eigenvector.
NOTE : The initial vector x0 is usually taken as a vector with
all components equal to 1.
Characteristic: The main advantage of this method is its
simplicity. And it can handle sparse matrices too large to store
as a full square array. Its disadvantage is its possibly slow
convergence.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

21. Eigen values and Eigen vectors by iteration
Power Method: Determining smallest eigenvalue.
If λ is the eigenvalue of A, then the reciprocal
1
λ
is the eigenvalue
of A−1.
The reciprocal of the largest eigenvalue of A−1 will be the
smallest eigenvalue of A.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

22. Example
Ex: Use power method to estimate the largest eigen value and the
corresponding eigen vector of A =
3 −5
−2 4
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

23. Example
Sol.: Let A =
3 −5
−2 4
and x0 =
1
1
Ax0 =
3 −5
−2 4
1
1
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

24. Example
Sol.: Let A =
3 −5
−2 4
and x0 =
1
1
Ax0 =
3 −5
−2 4
1
1
=
−2
2
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

25. Example
Sol.: Let A =
3 −5
−2 4
and x0 =
1
1
Ax0 =
3 −5
−2 4
1
1
=
−2
2
= −2
1
−1
= −2x1
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

26. Example
Sol.: Let A =
3 −5
−2 4
and x0 =
1
1
Ax0 =
3 −5
−2 4
1
1
=
−2
2
= −2
1
−1
= −2x1
Ax1 =
3 −5
−2 4
1
−1
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

27. Example
Sol.: Let A =
3 −5
−2 4
and x0 =
1
1
Ax0 =
3 −5
−2 4
1
1
=
−2
2
= −2
1
−1
= −2x1
Ax1 =
3 −5
−2 4
1
−1
=
8
−6
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

28. Example
Sol.: Let A =
3 −5
−2 4
and x0 =
1
1
Ax0 =
3 −5
−2 4
1
1
=
−2
2
= −2
1
−1
= −2x1
Ax1 =
3 −5
−2 4
1
−1
=
8
−6
= 8
1
−0.75
= 8x2
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

29. Example
Sol.: Let A =
3 −5
−2 4
and x0 =
1
1
Ax0 =
3 −5
−2 4
1
1
=
−2
2
= −2
1
−1
= −2x1
Ax1 =
3 −5
−2 4
1
−1
=
8
−6
= 8
1
−0.75
= 8x2
Ax2 =
3 −5
−2 4
1
−0.75
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

30. Example
Sol.: Let A =
3 −5
−2 4
and x0 =
1
1
Ax0 =
3 −5
−2 4
1
1
=
−2
2
= −2
1
−1
= −2x1
Ax1 =
3 −5
−2 4
1
−1
=
8
−6
= 8
1
−0.75
= 8x2
Ax2 =
3 −5
−2 4
1
−0.75
=
6.75
−5
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

31. Example
Sol.: Let A =
3 −5
−2 4
and x0 =
1
1
Ax0 =
3 −5
−2 4
1
1
=
−2
2
= −2
1
−1
= −2x1
Ax1 =
3 −5
−2 4
1
−1
=
8
−6
= 8
1
−0.75
= 8x2
Ax2 =
3 −5
−2 4
1
−0.75
=
6.75
−5
= 6.75
1
−0.7407
= 6.75x3
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

32. Example
Sol.: Let A =
3 −5
−2 4
and x0 =
1
1
Ax0 =
3 −5
−2 4
1
1
=
−2
2
= −2
1
−1
= −2x1
Ax1 =
3 −5
−2 4
1
−1
=
8
−6
= 8
1
−0.75
= 8x2
Ax2 =
3 −5
−2 4
1
−0.75
=
6.75
−5
= 6.75
1
−0.7407
= 6.75x3
∴ largest eigen value is 6.7015 and the corresponding eigen vector is
1
−0.7403
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

33. Example
Ex: Use power method to estimate the largest eigen value and the
corresponding eigen vector of A =
1 2
3 4
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

34. Example
Ex: Use power method to estimate the largest eigen value and the
corresponding eigen vector of A =
2 3
5 4
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

35. Example
Ex: Use power method to estimate the largest eigen value and the
corresponding eigen vector of A =
4 2
1 3
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

36. Example
Ex: Use power method to estimate the largest eigen value and the
corresponding eigen vector of A =


2 −1 0
−1 2 −1
0 −1 2


Dr. N. B. Vyas Numerical Methods Power Method for Eigen values

37. Example
Ex: Use power method to estimate the largest eigen value and the
corresponding eigen vector of A =


3 −1 0
−1 2 −1
0 −1 3


Dr. N. B. Vyas Numerical Methods Power Method for Eigen values