Singular Value Decomposition

Xinyi Zhao

doi:10.54254/2753-8818/2025.GL24952

Theoretical and Natural ScienceOpen access

Singular Value Decomposition

Research Article

Open Access

Singular Value Decomposition

Xinyi Zhao ^1*

¹ Institute of New York University, Brooklyn, United States

^*Corresponding author: xz3831@nyu.edu

Published on 20 July 2025

TNS Vol.125

ISSN (Print): 2753-8826

ISSN (Online): 2753-8818

ISBN (Print): 978-1-80590-233-1

ISBN (Online): 978-1-80590-234-8

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Abstract

Singular Value Decomposition (SVD) is a very important matrix factorization technique in linear algebra which generalizes the eigenvalue decomposition to both non square and non symmetric matrices. This report explains the theoretical foundation of SVD by numerical examples and the comparison of SVD with eigenvalue decomposition on the basis of versatility. Theoretical derivations, including proofs of SVD existence and uniqueness, are presented with practical implementation using Python. Experiments include image compression via truncated SVD and dimensional reduction on the Iris dataset. The results indicate that only top k singular values are enough to retain the essential data features and reduce storage requirements. For example, image compression with 50 singular values results in a MSE of 0.05 and visually clear images, as well as 4D data can be reduced into 2D without losing discriminative patterns. Findings confirm that SVD is computationally stable and efficient, and provides robust solution for rank approximation, noise reduction, and feature extraction. The study demonstrates that SVD's capability to break down data into intelligible components make it to remain relevant in big data analytics, scientific computation and artificial intelligence, with foreseeable future improvements also likely to increase its applications.

Keywords:

Matrices, Singular Value Decomposition, Rank, Orthogonal, Eigenvalue

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References

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References

[5]. Trefethen, L. N., & Bau, D. (2021). Numerical linear algebra. Society for Industrial and Applied Mathematics. https: //www.stat.uchicago.edu/~lekheng/courses/309/books/Trefethen-Bau.pdf

[6]. Lloyd, N. T., & David, B. (1997). Numerical linear algebra. Clinical Chemistry, iii edition, UK: Society for Industrial and Applied Mathematics.

[7]. Sauer, T. (2011). Numerical analysis. Addison-Wesley Publishing Company.

Cite this article

Zhao,X. (2025). Singular Value Decomposition. Theoretical and Natural Science,125,54-59.

Data availability

The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.

About volume

Volume title: Proceedings of CONF-APMM 2025 Symposium: Multi-Qubit Quantum Communication for Image Transmission over Error Prone Channels

ISBN: 978-1-80590-233-1(Print) / 978-1-80590-234-8(Online)

Editor: Anil Fernando

Conference website: https://2025.confapmm.org/

Conference date: 29 August 2025

Series: Theoretical and Natural Science

Volume number: Vol.125

ISSN: 2753-8818(Print) / 2753-8826(Online)

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