trace concept

Definition	The sum of a matrix's diagonal, which for a covariance is the total variance. Equation 0.3.
Example	diag(0.3, 1.7) has trace 2.0, the total variance.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `trace`, kind concept.
Status	measures [demonstrated]. Corrections: none recorded.
Defining equation	Book equation 0.3.
Assumptions and scope	The sum of a matrix’s diagonal. It is linear, the trace of the identity is the dimension, the trace of an outer product is the vector’s squared length, the trace of a product does not depend on the order, and the trace of a read operator times a rank-one error is the read distortion of that error. For a covariance the trace is the total variance, and the trace pairing of a read operator with a second moment is the invariant the ledger’s row OT-7 names, unchanged by a change of basis where the spectrum and effective rank are not.
Prior art	none recorded
Evidence	`geometric-observation/claims/LEDGER.md:30`, `lean/DataMiningAsObservation/Trace.lean`
Reviewed	not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page.

The sum of the diagonal, the total variance for a covariance.

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 0.10.

\[d_O=\operatorname{tr}(P_C\,M_\delta)=\mathbb E\!\left[\delta^{\top}P_C\,\delta\right],\qquad M_\delta=\mathbb E\!\left[\delta\delta^{\top}\right],\qquad P_C=I\ \Rightarrow\ d_O=\operatorname{tr}M_\delta.\]

Conditions

The sum of a matrix’s diagonal. It is linear, the trace of the identity is the dimension, the trace of an outer product is the vector’s squared length, the trace of a product does not depend on the order, and the trace of a read operator times a rank-one error is the read distortion of that error.
For a covariance the trace is the total variance, and the trace pairing of a read operator with a second moment is the invariant the ledger’s row OT-7 names, unchanged by a change of basis where the spectrum and effective rank are not.

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.

First stated

Chapter 0 section 0.2 of Data Mining as Observation, with the trace pairing of Volume 14’s invariance row OT-7.

Measurements

none

Failures and corrections

none

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/Trace.lean, theorems trace_add, trace_smul, trace_one, trace_vecMulVec, trace_mul_comm, trace_read, trace_identity_read, 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, 1, 3, 4, 13.

covariance matrix; read distortion; identity reader; effective rank.

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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