Power Iteration on Non-Normal Matrices: Spectral Radius, Jordan Effects, and Krylov-Based Accelerations

Abstract

The power iteration method is a foundational algorithm for approximating the dominant eigenvalue and eigenvector of matrices, widely applied in large-scale computations such as Google’s PageRank. While its convergence is well-understood for symmetric matrices, asymmetric (non-Hermitian) matrices present more complex challenges, influenced by the spectral radius and matrix structure. This paper investigates the mathematical relationship between the spectral radius and the convergence rate of power iteration in asymmetric matrices, focusing on both diagonalizable and non-diagonalizable cases. In diagonalizable matrices, convergence depends exponentially on the ratio r=|λ_2 |/|λ_1 |, where λ_1 and λ_2 are the dominant and subdominant eigenvalues. For non-diagonalizable matrices with Jordan blocks, additional polynomial factors k^(m-1) (where m is the block size) slow convergence, even with favorable eigenvalue gaps. Theoretical derivations, including matrix power computations in Jordan form, illustrate these effects. To address limitations such as slow convergence and inefficient information utilization, this paper extends the analysis to advanced iterative methods. Krylov subspace methods overcome single-vector restrictions by constructing optimal approximations in growing subspaces. The Arnoldi process generates orthogonal bases for non-symmetric matrices, leading to Hessenberg projections and Ritz approximations. Shift-and-invert techniques, based on spectral transformation theorems, accelerate convergence and target interior eigenvalues by manipulating the spectrum. Rayleigh quotient iteration introduces adaptive shifting, achieving cubic convergence for symmetric cases and quadratic for asymmetric. Subspace iteration generalizes to multiple eigenvalues using QR decomposition for block processing. Numerical experiments in Python compare basic power iteration with these extensions, demonstrating superior convergence in challenging scenarios (e.g., near-unity ratios or large Jordan blocks). Results highlight how these methods expand applicability in fields like Markov chains and control theory. This study underscores the need for structural awareness in eigenvalue computations and provides a framework for selecting appropriate extensions, enhancing efficiency in practical asymmetric matrix problems.