Chapter 6: Geometric Transformation and Eigen Decomposition. In the chapter mind map this icon labels Power Method. The discussion below is excerpted and lightly edited from § Power Method in Mathematics for AI and Machine Learning.

While the eigenvalue equation $A\mathbf{x} = \lambda\mathbf{x}$ provides a theoretical foundation, computing eigenvalues and eigenvectors for large matrices requires efficient numerical algorithms. The power method is an iterative algorithm that finds the dominant eigenvalue (the eigenvalue with the largest absolute value) and its corresponding eigenvector. Note that the absolute value of the dominant eigenvalue equals the spectral radius $\rho(A)$ (see Definition the referenced section, equation @eq:spectral-radius). This method is particularly useful when:

• Only the largest eigenvalue is needed (e.g., in Markov chains, the dominant eigenvalue determines the steady-state distribution)
• The matrix is too large for direct methods (e.g., computing $\det(A - \lambda I)$ is computationally prohibitive)
• Matrix-vector products $A\mathbf{v}$ are available but the full matrix decomposition is expensive

The power method exploits the fact that repeatedly applying $A$ to a random vector $\mathbf{v}$ will cause it to converge to the direction of the eigenvector corresponding to the dominant eigenvalue, since components along other eigenvectors decay exponentially relative to the dominant one.

This condition ensures that the dominant eigenvalue $\lambda_0$ is unique in magnitude, which guarantees convergence of the power method.