I need help properly implementing the power method to find the eigenvalues

I am struggling to develop a code that uses the power method to find the eigenvalues. The one that I have already is given is giving me eigenvalues that are extremely close to each other as shown below.

[Torsten]

The iter-loop is an iteration to approximate one eigenvalue, namely the largest. So it's not surprising that the outputs are close to each other - they are the iterates that converge to the largest eigenvalue.

[John D'Errico]

When you paste in a picture of your code, it is impossible to copy it in as text, so that we can test and run your code. When it is just as easy to insert text, why would you make it more difficult for people to help you?

Answer 1

[sai charan sampara]

Hello Brandon,

I understand that you are trying to find eigen vectors for a matrix using power method and want to know the reason behind getting nearly same vectors for different iterations.

The power method is an eigenvalue algorithm in which for a diagonalizable matrix, the algorithm will produce a number which is the greatest (in absolute value) eigenvalue of the matrix and an eigen vector corresponding to that largest eigen value. These are also known as principle eigen value and principle eigen vector. Hence the power method gives the eigen vector corresponding to the largest eigen value (absolute value) of the matrix.

In your case for multiple iterations, you will always get similar vectors because all these correspond to the principle eigen vector of the matrix. As the number of iterations increase the vector moves closer and closer to the principle eigen vector and hence the vectors are nearly the same. Power method can not give all the eigen vectors of a matrix.

You can refer to the below resources to learn more:

https://www.mathworks.com/matlabcentral/answers/7327-solved-power-method-eigenvalues