eigenvalue, eigenvector concept
| Definition | A direction a symmetric matrix only stretches, and the factor by which it stretches it. The eigenvectors of a covariance are its principal directions. Equation 0.5. Also eigenvalue, eigenvector, principal direction. |
|---|---|
| Example | diag(0.3, 1.7) has eigenvalues 0.3 and 1.7 with eigenvectors along the two axes. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id eigenvalue-eigenvector, kind concept. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | Book equation 0.5. |
| Assumptions and scope |
|
| Prior art | none recorded |
| Evidence | lean/DataMiningAsObservation/Eigen.lean |
| Reviewed | not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
Book equation 0.5.
\[\Sigma\,v_i=\lambda_i v_i,\qquad \Sigma=\sum_{i=1}^{d}\lambda_i\,v_i v_i^{\top},\qquad v_i\cdot v_j=0\ (i\ne j).\]
Book equation 0.7.
\[r_{\mathrm{eff}}=\frac{\big(\sum_i\lambda_i\big)^{2}}{\sum_i\lambda_i^{2}}.\]
Book equation 9.2.
\[N(\lambda)=\#\{k:\lambda_k\le\lambda\}\ \sim\ C_d\,\lambda^{d/2}\qquad\Rightarrow\qquad d=2\,\frac{d\log N}{d\log\lambda}.\]
Conditions
- A direction a symmetric matrix only stretches, and the factor by which it stretches it. Eigenvectors with distinct eigenvalues are orthogonal, every eigenvalue of a semidefinite matrix is nonnegative, and the quadratic form along an eigenvector is the eigenvalue times the squared length.
- The eigenvectors of a covariance are its principal directions, the basis in which chapter 4 pairs the covariance with the read operator, and the eigenvalues of a Laplacian are what the recognizer reads.
Conditions are curated in entries.toml rather than read
from a record.
Ledger
none
First stated
Chapter 0 section 0.4 of Data Mining as Observation, with the program’s spectrum records in readscope and the recognizer battery.
Measurements
none
Failures and corrections
none
Invariance envelope
none declared
Machine checked
lean/DataMiningAsObservation/Eigen.lean,
theorems pairing_symm, orthogonal_of_ne,
eigenvalue_nonneg, quad_eigen, at
observation-data-mining f3914f0; what the check covers is stated in the
book’s appendix
C.
Used in
Data Mining as Observation primer L, chapters 0, 3, 4, 9, 11.
Related
covariance matrix; effective rank; whitening; recognizer; Laplacian.
See also
Ledger rows that cite the entry’s records without naming it: OT-7.
Sources-table rows that share a record with the entry without naming it: chapter 4 section 4.2, chapter 9 section 9.4, chapter 14 section 14.3, chapter 14 section 14.5, chapter 14 section 14.7.
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.