{"id":"eigenvalue-and-eigenvector","domain":"algebra","type":"Concept","title":"Eigenvalue & Eigenvector","slug":"eigenvalue-and-eigenvector","url":"/mathematics/algebra/eigenvalue-and-eigenvector/","summary":"Eigenvectors are directions a matrix only scales, never rotates; eigenvalues are the scaling factors. Basis of PageRank, quantum mechanics, vibration analysis.","formal_definition":"Av = lambda*v","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Strang, G. (2016). Introduction to Linear Algebra. 5th ed."},{"tier":1,"citation":"Horn, R.A. and Johnson, C.R. (2012). Matrix Analysis. 2nd ed."},{"tier":2,"citation":"Page, L., Brin, S., Motwani, R. and Winograd, T. (1999). The PageRank Citation Ranking. Stanford InfoLab."},{"tier":3,"citation":"Strogatz, S. (2019). Infinite Powers. Houghton Mifflin Harcourt."}],"relationships":[{"type":"computed_via","target":"determinant","label":"Determinant — characteristic polynomial"},{"type":"uses","target":"matrix","label":"Matrix"},{"type":"uses","target":"complex-number","label":"Complex Number"},{"type":"related_to","target":"normal-distribution","label":"Normal Distribution — covariance"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.8"}