covariance matrix concept

Definition	The matrix of pairwise covariances of a set of rows, whose trace is the total variance. Equation 0.3. Also covariance.
Example	Two columns with variances 0.3 and 1.7 and no covariance have covariance matrix diag(0.3, 1.7) and trace 2.0.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `covariance-matrix`, kind concept.
Status	measures [demonstrated]; measures [predicted]. Corrections: none recorded.
Defining equation	Book equation 0.3.
Assumptions and scope	The weighted average of the centred rows’ outer products, the same construction as the read operator with the centred row in place of the sensitivity. Its quadratic form along a unit direction is the variance of the projection onto it, its trace is the total variance, and it is positive semidefinite. Chapter 4 pairs it with the read operator in one basis. Where the two are proportional there is no flip, and where they are not the water-filling allocation reads both.
Prior art	none recorded
Evidence	`geometric-observation/claims/LEDGER.md:30`, `geometric-observation/claims/LEDGER.md:33`, `lean/DataMiningAsObservation/Covariance.lean`
Reviewed	not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page.

Variances on the diagonal and covariances off it, symmetric and positive semidefinite.

Equation

Book equation 0.3.

\[\Sigma_{ij}=\mathbb E\big[(x_i-\mu_i)(x_j-\mu_j)\big],\qquad \operatorname{tr}\Sigma=\sum_{i}\Sigma_{ii}.\]

Book equation 0.5.

\[\Sigma\,v_i=\lambda_i v_i,\qquad \Sigma=\sum_{i=1}^{d}\lambda_i\,v_i v_i^{\top},\qquad v_i\cdot v_j=0\ (i\ne j).\]

Conditions

The weighted average of the centred rows’ outer products, the same construction as the read operator with the centred row in place of the sensitivity. Its quadratic form along a unit direction is the variance of the projection onto it, its trace is the total variance, and it is positive semidefinite.
Chapter 4 pairs it with the read operator in one basis. Where the two are proportional there is no flip, and where they are not the water-filling allocation reads both.

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 under P' = 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.
measures. OT-2 [predicted]. Loading is a covariance, not a distance: reading error under a change of measure is priced by ε·‖E[h·A]‖ — predicted from the base measure alone — and a full-magnitude shift orthogonal to the operator’s variation does nothing. geometric-observation/claims/LEDGER.md:33.

First stated

Chapter 0 section 0.2 of Data Mining as Observation, paired with the read operator in chapter 4, and Volume 14 chapter 5, geometric-observation/chapters/ch05_the_read_metric_and_the_quotient.md:7-48.

Measurements

none

Failures and corrections

none

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/Covariance.lean, theorems var_eq_quad, trace_eq_total, quad_nonneg, cov_symm, 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, 9, 10, 11, 12.

read operator; whitening; water-filling; alignment; explained variance.

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.

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