The Power Iteration Method is an iterative algorithm for finding the *principal eigenvector* of a matrix $L$, which is the [[Eigenvectors|eigenvector]] corresponding to the largest eigenvalue in magnitude. **Idea:** Instead of computing all eigenvectors (a computationally expensive process), the Power Iteration Method focuses on iteratively approximating the principal eigenvector. Over successive iterations, the influence of the largest eigenvalue dominates, causing the algorithm to converge to the principal eigenvector. **Algorithm:** 1. *Initialization:* Start with a random initial vector $r_0$. 2. *Iterative multiplication:* Multiply the matrix $L$ by the current vector $r_i$ to produce a new vector $r_{i+1}$. $ r_{i+1} = Lr_i$ 3. *Normalization:* Normalize $r_{i+1}$ by its [[Vector Length|length]] after each iteration to prevent it from growing arbitrarily large. $ r_{i+1}= \frac{r_{i+1}}{\vert \vert r_{i+1} \vert \vert}$ 4. *Convergence:* Repeat the process until $r_{i+1}$ converges to $r_i$ (i.e. $r_{i+1} \approx r_i$). At convergence, we effectively identified a vector $r$ that remains unchanged after being transformed by a matrix $L$. This satisfies the property of an eigenvector whose corresponding eigenvalue is $\lambda=1$. $ \begin{align}Ax = \lambda x \\ Lr = 1r \end{align}$