spectrum concept

Definition	The list of a matrix's eigenvalues. Chapter 0 section 0.4. Also spectra.
Example	diag(0.3, 1.7) has spectrum (0.3, 1.7), trace 2.0, and effective rank 1.34.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `spectrum`, kind concept.
Status	measures [demonstrated]. Corrections: none recorded.
Defining equation	Book equation 0.5.
Assumptions and scope	The list of a matrix’s eigenvalues. Its effective rank, the square of its sum over the sum of its squares, lies between one and the dimension, is the dimension for a flat spectrum, and is one for a single eigenvalue. The spectrum is not invariant under a change of basis of the input where the trace pairing and the rank are, and the Laplacian’s spectrum is what the recognizer reads for dimension and shape.
Prior art	none recorded
Evidence	`geometric-observation/claims/LEDGER.md:30`, `lean/DataMiningAsObservation/EffectiveRank.lean`
Reviewed	not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page.

The eigenvalues of a matrix.

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.

\[r_{\mathrm{eff}}=\frac{\big(\sum_i\lambda_i\big)^{2}}{\sum_i\lambda_i^{2}}.\]

Conditions

The list of a matrix’s eigenvalues. Its effective rank, the square of its sum over the sum of its squares, lies between one and the dimension, is the dimension for a flat spectrum, and is one for a single eigenvalue.
The spectrum is not invariant under a change of basis of the input where the trace pairing and the rank are, and the Laplacian’s spectrum is what the recognizer reads for dimension and shape.

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 0.4 of Data Mining as Observation, with the program’s spectrum records in readscope, readscope/readscope/spectrum.py:35-70, and the recognizer battery.

Measurements

none

Failures and corrections

none

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/EffectiveRank.lean, theorems sq_sum_le, effRank_le, sum_sq_le_sq_sum, one_le_effRank, effRank_const, effRank_single, 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, 14.

eigenvalue, eigenvector; effective rank; recognizer; Laplacian; isotropic.

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.

← spectral embeddingsplit →