spectral decomposition concept
| Definition | A symmetric matrix written as the sum of its eigenvalues times the outer products of its orthonormal eigenvectors. Powers, inverses, and square roots act on the eigenvalues alone, and the quadratic form becomes a weighted sum of squared coordinates in the eigenvector basis. Primer L, equations L.14 and L.15. |
|---|---|
| Example | Rows (2, 1) and (1, 2) equal 3 times the matrix with every entry one half plus 1 times the matrix with rows (1/2, -1/2) and (-1/2, 1/2). |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id spectral-decomposition, kind concept. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | Book equation L.14. |
| Assumptions and scope | none |
| Prior art | none recorded |
| Evidence | none |
| Reviewed | semantic review 2026-09-09; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
Book equation L.14.
\[A=\sum_{i=1}^{d}\lambda_i\,v_iv_i^{\top}=V\Lambda V^{\top},\qquad V^{\top}V=I,\qquad \Lambda=\operatorname{diag}(\lambda_1,\dots,\lambda_d).\]
Conditions
none
Ledger
none
First stated
Primer L section L.7 of Data Mining as Observation, added in draft 0.3 (2026-09-09) for the ECE 514 readers whose first courses are far behind. The idea is standard and TSK Appendix A covers it at length.
Measurements
none
Failures and corrections
none
Invariance envelope
none declared
Machine checked
none
Used in
Data Mining as Observation primer L.
Related
eigenvalue, eigenvector; outer product; whitening; spectrum; symmetric matrix.
See also
Book equations stated beside the entry’s terms, not defining it: L.15.
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.