Penerapan Metode Singular Value Decomposition (SVD) dalam Menyelesaikan Sistem Persamaan Linear
DOI:
https://doi.org/10.53696/venn.v5i1.445Keywords:
Singular Value Decompsition (SVD), Basis Ortonormal, Nilai Singular, Sistem Persamaan Linear (SPL)Abstract
Eliminasi Gauss atau eliminasi Gauss-Jordan merupakan metode yang sangat lazim dan banyak digunakan untuk menentukan solusi dari sistem persamaan linear Ax = b. Namun demikian metode ini hanya dapat digunakan untuk sistem persamaan linear yang konsisten. Sehingga diperlukan suatu metode yang tidak hanya dapat digunakan untuk menentukan solusi dari suatu sistem persamaan linear konsisten tapi juga dapat memberikan solusi untuk sistem persamaan linear yang tidak konsisten. Solusi aproksimasi untuk sistem persamaan linear yang tidak konsisten dapat ditentukan dengan menggunakan metode Singular Value Decomposition (SVD). Metode SVD ini adalah metode faktorisasi matriks yang berkaitan erat dengan nilai singular matriks. Analisis SVD dapat digunakan untuk menentukan suatu solusi pendekatan dengan memanfaatkan basis ortonormal untuk empat subruang fundamental yang terkait dengan matriks A yaitu R(A), R(AH), N(A) dan N(AH) . Basis-basis tersebut dapat digunakan untuk menghitung solusi sistem persamaan linear yang konsisten dan tidak konsisten.
Downloads
References
Akritas, A. G., Malaschonok, G. I., & Vigklas, P. S. (2006). The SVD-Fundamental Theorem of Linear Algebra. Nonlinear Analysis: Modelling and Control, 11(2), 123–136. https://doi.org/10.15388/NA.2006.11.2.14753
Andrilli, S., & Hecker, D. (2022). Elementary linear algebra. Academic Press.
Anton, H., & Rorres, C. (2013). Elementary linear algebra: applications version. John Wiley & Sons.
Aryani, F., & Yulianti, D. (2014). Aplikasi Metode Singular Value Decomposition (SVD) pada Sistem Persamaan Linier Kompleks. SITEKIN: Jurnal Sains, Teknologi Dan Industri, 10(1), 67–76. https://doi.org/10.24014/SITEKIN.V10I1.577
Ayres, F. (1982). Theory and Problem of Matriks. Mc Graw-Hill, Singapura.
Bentbib, A. H., & Kanber, A. (2015). Block power method for SVD decomposition. Analele Stiintifice Ale Universitatii Ovidius Constanta, Seria Matematica, 23(2), 45–58.
Dembélé, D. (2026). A power method for computing singular value decomposition. Journal of Computational and Applied Mathematics, 484, 117451. https://doi.org/10.1016/j.cam.2026.117451
Haidir Ahmad, I., & Ratnasari, L. (2010). Menyelesaikan Sistem Persamaan Linier Menggunakan Analisis Svd. Jurnal Matematika, 13, No.1(1), 40–45.
Kalman, D. (1996). A Singularly Valuable Decomposition: The SVD of a Matrix. The College Mathematics Journal, 27(1), 2–23. https://doi.org/10.1080/07468342.1996.11973744
Larson, R., Edwards, B. H., & Falvo, D. C. (2013). Elementary linear algebra.
Marzuki, C. C., Hariati, D., Afmilda, J., Husna, N., & Nanda, P. (2018). Penyelesaian Sistem Persamaan Linier Fully Fuzzy Menggunakan Metode Dekomposisi Nilai Singular (SVD). Jurnal Matematika MANTIK, 4(2), 143–149. https://doi.org/doi.org/10.15642/mantik.2018.4.2.143-149
Meng, K., & Li, S. (2026). Fast one-step multi-view clustering via low-rank singular value decomposition. Neurocomputing, 678, 133154. https://doi.org/10.1016/j.neucom.2026.133154
Nicholson, W. Keith. (2001). Elementary linear algebra.
Sutojo, T., Nurhadiono, B., Astuti, E. Z., Astuti, S., Rahayu, Y., & Mulyanto, E. (2010). Teori dan aplikasi aljabar linear dan matriks.
Wibowo, E. S. A., Yulida, Y., & Thresye, T. (2017). Penyelesaian Sistem Persamaan Diferensial Linier Melalui Diagonalisasi Matriks. EPSILON: JURNAL MATEMATIKA MURNI DAN TERAPAN (EPSILON: JOURNAL OF PURE AND APPLIED MATHEMATICS), 7(2), 12–19. https://doi.org/10.20527/EPSILON.V7I2.95
Zhu, Q., Wu, S., Xie, S., Yin, Z., & Agaian, S. (2026). Nested quaternion matrix and recursive approach to its singular value decomposition. Journal of Computational and Applied Mathematics, 485, 117543. https://doi.org/10.1016/j.cam.2026.117543
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2026 Asmaun Asmaun
This work is licensed under a Creative Commons Attribution 4.0 International License.
The authors certify that:
- if the manuscript is co-authored, they are authorized by their co-authors to enter into these arrangements.
- the work described has not been formally published before in a registered ISSN or ISBN media, except in the form of an abstract or as part of a published lecture, review, or thesis.
- it is not under consideration for publication elsewhere,
- its publication has been approved by all the author(s) and by the responsible authorities – tacitly or explicitly – of the institutes where the work has been carried out.
- they secure the right to reproduce any material that has already been published or copyrighted elsewhere (it does not infringe on the rights of others).
- they agree to the license and copyright agreement.
All articles published are licensed under Creative Commons Attribution 4.0 International License.
- Authors retain copyright and other proprietary rights related to the article.
- Authors retain the right and are permitted to use the substance of the article in their own future works, including lectures and books.
- Authors grant the journal right of first publication with the work simultaneously licensed under Creative Commons Attribution License (CC BY 4.0) that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal.
- Authors are permitted and encouraged to post or self-archive their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.
