Mathematical Applications of Eigenvalues and Eigenvectors: A Comprehensive Review

Eigenvalues and eigenvectors are some of the most useful and commonly used ideas in linear algebra, and they play a key role in math, science, and engineering today. Originating from the work of Euler and Cauchy in the eighteenth and nineteenth centuries and formalized through the development of linear algebra and spectral theory, the eigenvalue problem—expressed as Aν = λν, where A is a square matrix, ν is the eigenvector, and λ is the corresponding eigenvalue—underpins a diverse spectrum of theoretical and applied disciplines. This review paper presents a comprehensive survey of the mathematical foundations and principal application domains of eigenvalues and eigenvectors, synthesizing findings from more than 35 published sources spanning 1877 to 2024. The review covers: the formal mathematical definition and derivation of eigenvalues and eigenvectors via the characteristic equation; methods of computation including the power method and QR algorithm; and major application domains including structural engineering vibration analysis (modal analysis), Principal Component Analysis (PCA) for dimensionality reduction in data science and machine learning, Google’s PageRank algorithm for web page ranking, quantum mechanics energy level determination, image compression through singular value decomposition (SVD), control systems stability analysis, facial recognition through eigenfaces, Markov chain steady-state analysis, and graph theory. Worked numerical examples are provided for the characteristic equation method, PCA covariance matrix decomposition, and modal analysis of a two-degree-of-freedom mass-spring system. The review demonstrates that eigenvalues and eigenvectors transcend disciplinary boundaries, serving as a unifying mathematical language across physics, engineering, computer science, economics, and biology. Research trends including sparse eigenvalue decomposition, eigenvector-based deep learning architectures, and randomized numerical methods for large-scale eigenproblems are identified as priority directions for future investigation.