Finding eigenvalues with the power series method

[Ready2GoXtr]

Homework Statement

Consider the matrix [1,-5,5;-3,-1,3;1,-2,2]
Do four interations of the power method, beginning at [1,1,1] to approximate the dominant eigenvalues of A

Homework Equations

Matrix multiplication

The Attempt at a Solution

Okay my issue with this problem is this
I set V₀ = [1;1;1],
Now I go to calculate u₁
u₁ = A*V₀ = [1,-5,5;-3,-1,3;1,-2,2]*[1;1;1]=[1;-1;1], V₁ = u₁ / (?) and what value should i divide it by, which one has the largest magnitude, would it be -1, because I know that is unique, otherwise, is it 1?
Because I've tried both ways and I am not sure which way to go on this.

 

[Redbelly98]

You divide by the magnitude (i.e. length) of u1, or |u1|.

 

[Architeuthis Dux]

it is important to consider all possible solutions and approaches to a problem. In this case, the power series method is a valid approach to finding eigenvalues of a matrix. However, it is important to note that there are other methods such as the QR algorithm or the Jacobi method that may also be useful in finding eigenvalues.

In regards to the specific problem given, it is important to clarify the steps of the power method. The first step is to choose an initial vector, which in this case is V0 = [1;1;1]. Next, we calculate u1 = A*V0 = [1,-5,5;-3,-1,3;1,-2,2]*[1;1;1]=[1;-1;1]. The next step is to normalize u1 by dividing it by its largest magnitude element, which in this case is -1. So V1 = u1 / (-1) = [-1;1;-1].

For the next iteration, we repeat the process by calculating u2 = A*V1 and normalizing it to get V2. This process is repeated for four iterations to approximate the dominant eigenvalues of A. It is important to note that the power method may not always converge to the dominant eigenvalues and the number of iterations needed may vary depending on the matrix.

In summary, the power series method is a valid approach to finding eigenvalues, but it is important to carefully follow the steps and consider other methods as well.