singular value concept
| Definition | The size of one rank-one term of the singular value decomposition, the square root of an eigenvalue of A transposed A. The count of nonzero ones is the rank, and their squares sum to the squared Frobenius norm. Primer L, equations L.17 and L.18. |
|---|---|
| Example | Rows (1, 2) and (2, 4) have singular values 5 and 0, so the rank is one. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id singular-value, kind concept. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | Book equation L.17. |
| 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.17.
\[A^{\top}A\,v_i=\sigma_i^{2}\,v_i,\qquad u_i=\frac{Av_i}{\sigma_i},\qquad \Sigma=\frac{A^{\top}A}{n}\ \text{for centered } A.\]
Book equation L.18.
\[\|A\|_F^{2}=\sum_{i,j}A_{ij}^{2}=\sum_i\sigma_i^{2},\qquad \min_{\operatorname{rank}B\le k}\|A-B\|_F^{2}=\sum_{i>k}\sigma_i^{2}.\]
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, 9.
Related
singular value decomposition; rank; Frobenius norm; spectrum.
See also
none
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.