Comparative Analysis of Singular Value Decomposition and Cholesky Decomposition Methods for Solving Large Scale Linear Systems

This study presents a comparative analysis of matrix decomposition methods for solving large-scale linear systems, focusing on computational efficiency, numerical stability, and applicability across different problem domains. Matrix decomposition is fundamental in numerical linear algebra, as it simplifies complex systems into forms that are easier to solve. The study reviews classical methods of SVD and Cholesky decomposition, highlighting their strengths and limitations. SVD extend decomposition to spectral analysis and dimensionality reduction, making them valuable in machine learning and data science. Cholesky decomposition, in contrast, offers speed and stability in positive-definite systems. Applications in optimization, artificial intelligence, recommender systems and structural engineering are examined to demonstrate practical relevance. The findings show that method performance is problem-dependent,with trade-offs between speed, accuracy, and scalability. The study concludes that although classical methods remain foundational, modern large-scale and sparse problems require advances such as randomized and hybrid decomposition techniques