Power Iteration

Implement the power iteration method to find the dominant eigenvector of a matrix. This task involves understanding the underlying concept of eigenvalues and eigenvectors, which are crucial in linear algebra and have numerous applications in computer vision and machine learning.

The power iteration method is an iterative algorithm that finds the eigenvector corresponding to the largest eigenvalue of a matrix $A$. The process involves starting with a random vector $\mathbf{v}_0$ and iteratively updating it using the matrix $A$. The key idea is to repeatedly apply the matrix $A$ to the current vector and normalize the result.

Here are the general steps:

1. Initialize a random vector $\mathbf{v}_0$
2. Iterate: update the vector using the matrix $A$ and normalize the result
3. Repeat until convergence

This technique is widely used in principal component analysis (PCA).