singular value decomposition concept
| Definition | Any matrix written as a sum of rank-one outer products ordered by size, with orthonormal vectors on both sides, the eigenvector idea for a matrix that is not square. Its right singular vectors are the eigenvectors of A transposed A, and for centered data they are the principal components. Primer L, equations L.16 and L.17. |
|---|---|
| Example | Rows (1, 1), (1, 0), and (0, 1) have A transposed A with rows (2, 1) and (1, 2), so the singular values are root 3 and 1. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id singular-value-decomposition, kind concept. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | Book equation L.16. |
| Assumptions and scope | none |
| Prior art | none recorded |
| Evidence | none |
| Reviewed | semantic review 2026-09-09; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
Book equation L.16.
\[A=\sum_{i=1}^{r}\sigma_i\,u_iv_i^{\top}=U\Sigma_{\!s}V^{\top},\qquad \sigma_1\ge\sigma_2\ge\cdots\ge\sigma_r>0,\qquad U^{\top}U=I,\ V^{\top}V=I.\]
Conditions
none
Ledger
none
First stated
Primer L section L.8 of Data Mining as Observation, added in draft 0.3 (2026-09-09) for the ECE 514 readers whose first courses are far behind. The idea is standard and TSK Appendix A covers it at length.
Measurements
none
Failures and corrections
none
Invariance envelope
none declared
Machine checked
none
Used in
Data Mining as Observation primer L, chapters 0.
Related
singular value; principal component analysis; low-rank approximation; latent semantic analysis; eigenvalue, eigenvector.
See also
Book equations stated beside the entry’s terms, not defining it: L.17.
Status
Generated 2026-09-10 by encyclopedia/generate.py; book
at observation-data-mining f3914f0; the commit of every record is listed
in the encyclopedia’s provenance.