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The Approximation of One Matrix by Another of Lower Rank
Abstract
The mathematical problem of approximating one matrix by another of lower rank is closely related to the fundamental postulate of factor-theory. When formulated as a least-squares problem, the normal equations cannot be immediately written down, since the elements of the approximate matrix are not independent of one another. The solution of the problem is simplified by first expressing the matrices in a canonic form. It is found that the problem always has a solution which is usually unique. Several conclusions can be drawn from the form of this solution. A hypothetical interpretation of the canonic components of a score matrix is discussed.
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Citations
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“This is also why truncated SVD is understood as the best low-rank approximation as it preserves the information most compactly [Eckart & Young, 1936].”
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“According to the Eckart-Young Theorem [42], the best-possible low-rank approximation of matrix M obtained by minimizing the Frobenius norm of the difference between M and the approximated matrix ˜ M is achieved through the truncated SVD of M .”
MethodShinwari, H. U. K., & Usama, M. (2025) · arXiv · doi:10.48550/arxiv.2506.18267 · Cited by 3
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Metadata
- Authors
- Eckart, C., & Young, G.
- Venue
- Psychometrika
- Publisher
- Cambridge University Press (CUP)
- Published
- 1936-09
- Volume
- 1
- Issue
- 3
- Pages
- 211-218
- ISSN
- 0033-3123 (ISSN-L)
- 1860-0980
- OpenAlex
- W2004026774
- Cited by
- 3,800
- References
- 0
- FWCI
- 6.11
- Type
- journal article
- Language
- en
- Open access
- closed
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