On Singular Value Decomposition

Abstract

This study investigates the algebraic and geometric structures underlying Singular Value Decomposition (SVD), one of the most fundamental matrix factorizations in linear algebra with extensive applications in both theoretical and applied mathematics. The research aims to define SVD and derive its formal construction, identify the algebraic properties and conditions that guarantee its existence and uniqueness, provide a geometric interpretation of SVD through orthogonal transformations and ellipsoids, and demonstrate how its mathematical structure contributes to solving practical problems, particularly in data analysis. Employing a theoretical research approach, the study establishes the necessary foundations in linear algebra, including orthogonality, eigenvalues, and the Spectral Theorem, and presents a constructive proof of the existence of SVD for any real matrix by analyzing the symmetric positive semi-definite matrix ( ). The study further illustrates the geometric interpretation of SVD as a sequence of rotation, scaling, and rotation transformations that map the unit circle or sphere into an ellipsoid. To demonstrate its practical relevance, a computational example involving student grade data is analyzed using SVD. The findings show that the existence of SVD is guaranteed for every real matrix, with singular values being unique when arranged in descending order, while singular vectors may not be unique in the presence of repeated singular values. Geometrically, SVD is shown to represent linear transformations through orthogonal rotations and principal-axis scaling, providing a clear visualization of how matrices transform space. Moreover, the decomposition effectively identifies dominant and secondary patterns in student performance data by separating the data into left singular vectors, singular values, and right singular vectors, thereby revealing underlying latent structures. The study concludes that SVD serves as a powerful bridge between algebraic theory and geometric intuition, explaining concepts such as rank and dimensionality through the mapping of a unit sphere to an ellipsoid. Furthermore, its rank-one matrix decomposition forms the theoretical basis for low-rank approximation, making SVD an indispensable mathematical tool for pattern extraction, dimensionality reduction, noise filtering, and modern computational analysis.