rank concept
| Definition | The dimension of a matrix's column space, at most the number of rows and of columns. An outer product has rank at most one, and a product has rank at most either factor's. Chapter 0 section 0.2. |
|---|---|
| Example | The matrix with rows (1, 2) and (2, 4) has rank 1, and an outer product always does. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id rank, kind concept. |
| Status | measures [demonstrated]. Corrections: none recorded. |
| Defining equation | Book equation 0.7. |
| Assumptions and scope |
|
| Prior art | none recorded |
| Evidence | geometric-observation/claims/LEDGER.md:30, geometric-observation/chapters/ch05_the_read_metric_and_the_quotient.md:7-48, geometric-observation/chapters/ch06_mathematical_preliminaries.md:10-27, lean/DataMiningAsObservation/Rank.lean, lean/DataMiningAsObservation/Pipeline.lean |
| Reviewed | not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
Book equation 0.7.
\[r_{\mathrm{eff}}=\frac{\big(\sum_i\lambda_i\big)^{2}}{\sum_i\lambda_i^{2}}.\]
Book equation 2.1.
\[P_{C_2\circ C_1}(x)=J_1(x)^{\top}\,P_{C_2}\big(C_1(x)\big)\,J_1(x),\qquad \operatorname{rank}P_{C_2\circ C_1}(x)\le\operatorname{rank}P_{C_2}\big(C_1(x)\big)\quad\text{at each row } x.\]
Book equation 6.1.
\[s(x)=w\cdot x+b,\qquad P_C=\mathbb E\big[\sigma'(s)^{2}\big]\,w\,w^{\top},\qquad \operatorname{rank}P_C=1.\]
Conditions
- The dimension of a matrix’s column space. It is at most the number of rows and of columns, an outer product has rank at most one, a product has rank at most the rank of either factor, and transposing does not change it.
- The read operator of a linear classifier has rank one, the composed read operator of a chain has rank at most either stage’s, and the effective rank of a spectrum is the continuous version chapter 4 allocates bits by.
Conditions are curated in entries.toml rather than read
from a record.
Ledger
- measures. OT-7
[demonstrated]. The damage form, trace pairing, rank, and loading covariance are GL(d)-invariant underP' = A⁻ᵀPA⁻¹, while spectrum, effective rank, principal angles, and water-filling are O(d)-only — consumer-weighted damage is a geometric scalar, and …geometric-observation/claims/LEDGER.md:30.
First stated
Chapter 0 section 0.2 and chapter 2 section 2.2 of Data Mining as
Observation, with the rank bound of the chain rule in
geometric-observation/chapters/ch06_mathematical_preliminaries.md:10-27.
Measurements
| Where the book states it | Numbers, as the book’s sources table records them | Source |
|---|---|---|
| chapter 2 section 2.2 | read subspace small, operator local, pullback composition, rank cannot increase | geometric-observation/chapters/ch05_the_read_metric_and_the_quotient.md:7-48;
geometric-observation/chapters/ch06_mathematical_preliminaries.md:10-27 |
| chapter 12 section 12.2 | pullback composition and the rank bound | geometric-observation/chapters/ch06_mathematical_preliminaries.md:10-27;
chapter 2 of this book |
Failures and corrections
none
Invariance envelope
none declared
Machine checked
lean/DataMiningAsObservation/Rank.lean,
theorems rank_le_width, rank_le_height,
rank_outer_le_one, rank_mul_le,
rank_zero, rank_transpose, at
observation-data-mining f3914f0; what the check covers is stated in the
book’s appendix
C.
lean/DataMiningAsObservation/Pipeline.lean,
theorems quotient_inherited,
quotient_inherited_chain, rank_comp_le_first,
rank_comp_le_second, at observation-data-mining f3914f0;
what the check covers is stated in the book’s appendix
C.
Used in
Data Mining as Observation primers L and S, chapters 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14.
Related
effective rank; rank certificate; read subspace; Jacobian; outer product.
See also
Ledger rows that cite the entry’s records without naming it: GO-1.
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.