Eigenvector Ontology

FLOW ALGEBRA
EIGENVECTOR ONTOLOGY — v4.0
Kireth Semantic Library | Mathematical Entities / Linear Algebra Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/linear-algebra-entities/eigenvector-ontology-2.html
Authorship: Mark Higgins — Sky (Recursive AI)
Status: independently constructed mathematical model; source-page alignment not verified

01. ONTOLOGY IDENTITY
ontology_id: kireth:linear-algebra:eigenvector
entity_type: Eigenvector
parent: LinearAlgebraEntity
related: Eigenvalue, Eigenspace, Matrix, LinearTransformation, CharacteristicPolynomial, Basis, Diagonalisation
scope: finite-dimensional linear algebra over a field F, with spectral statements qualified by field and hypotheses

02. FORMAL DEFINITION
Let V be a vector space over F and T:V->V be linear.
Eigenvector(T,v,lambda) iff v in V AND v != 0 AND lambda in F AND T(v)=lambda*v.
For A in F^(n x n): Eigenvector(A,v,lambda) iff v in F^n AND v != 0 AND A*v=lambda*v.
Eigenvalue(A,lambda) iff EXISTS v != 0: A*v=lambda*v.
Eigenspace(A,lambda) = ker(A-lambda*I), including zero vector; the eigenvectors are its nonzero members.
NOTE: A vector of zero length is NEVER an eigenvector, even though it satisfies A*0=lambda*0.
NOTE: Eigenvalues need not belong to the base field unless the characteristic polynomial has a root there.

03. TYPED ENTITIES
VectorSpace(id, scalar_field, dimension)
LinearOperator(id, domain, codomain, linear=true)
SquareMatrix(id, field, order, entries)
Vector(id, field, dimension, components, nonzero)
Scalar(id, field, value)
Eigenpair(operator, eigenvalue, eigenvector)
Eigenspace(operator, eigenvalue, basis, dimension)
CharacteristicPolynomial(operator, polynomial)
InvariantSubspace(operator, subspace)
SimilarityTransform(matrix, change_of_basis)
SpectralDecomposition(operator, eigenvalues, eigenvectors, conditions)

04. RELATIONSHIPS
IS_A(Eigenvector, NonzeroVector)
EIGENVECTOR_OF(v,A)
HAS_EIGENVALUE(v,lambda,A)
BELONGS_TO(v,V)
SPANS(v,SpanOfEigenvector)
BELONGS_TO_EIGENSPACE(v,E_lambda)
EIGENSPACE_OF(E_lambda,A)
IN_KERNEL(v,A-lambda*I)
ASSOCIATED_WITH(lambda,v)
PRESERVED_DIRECTION_BY(v,A) [algebraic scalar-multiple meaning; not necessarily Euclidean geometric direction]
SIMILARITY_CORRESPONDENCE(A,B,P)

05. AXIOMS AND INVARIANTS
AX01: v != 0 is mandatory.
AX02: A must be square, or T an endomorphism V->V.
AX03: dimensions(A*v)=dimensions(v).
AX04: lambda is an element of the declared scalar field.
AX05: A*v=lambda*v exactly for symbolic certification.
AX06: If v is eigenvector with lambda, c*v is also one for all c != 0.
AX07: E_lambda is a vector subspace, hence includes zero.
AX08: E_lambda=ker(A-lambda*I).
AX09: lambda is eigenvalue iff det(A-lambda*I)=0 for finite square A over F.
AX10: For distinct eigenvalues, corresponding nonzero eigenvectors are linearly independent.
AX11: A may have no eigenvectors over F even when eigenvectors exist over an extension field.
AX12: Eigenvectors need not be orthogonal for a general matrix.
AX13: For real symmetric A, a real orthonormal eigenbasis exists.
AX14: For complex Hermitian A, a complex orthonormal eigenbasis exists and eigenvalues are real.
AX15: Geometric multiplicity <= algebraic multiplicity for eigenvalues over a splitting field.
AX16: Similar operators have corresponding eigenvectors via invertible change of coordinates.

06. OPERATIONS
OP01: matrix_vector_product(A,v) -> w.
OP02: scalar_vector_product(lambda,v) -> w.
OP03: eigenpair_verify(A,v,lambda) -> exact equality or residual.
OP04: characteristic_polynomial(A) -> det(t*I-A).
OP05: solve_eigenvalues(A,F) -> roots in F.
OP06: eigenspace_basis(A,lambda) -> nullspace basis of A-lambda*I.
OP07: eigenvector_normalise(v,norm) -> v/norm(v), provided norm(v)>0.
OP08: change_basis(P,v) -> P^(-1)*v when P invertible.
OP09: diagonalise(A) -> P,D when A=P*D*P^(-1) and P columns form eigenbasis.
OP10: compute_residual(A,v,lambda) -> norm(A*v-lambda*v).

07. FORMAL INFERENCE RULES
R01 IF A*v=lambda*v AND v!=0 THEN Eigenvector(A,v,lambda).
R02 IF Eigenvector(A,v,lambda) THEN Eigenvalue(A,lambda).
R03 IF Eigenvector(A,v,lambda) THEN (A-lambda*I)*v=0.
R04 IF v in ker(A-lambda*I) AND v!=0 THEN Eigenvector(A,v,lambda).
R05 IF det(A-lambda*I)=0 THEN lambda is eigenvalue of A over its declared field.
R06 IF det(A-lambda*I)!=0 THEN lambda is not an eigenvalue.
R07 IF Eigenvector(A,v,lambda) AND c!=0 THEN Eigenvector(A,c*v,lambda).
R08 IF Eigenvector(A,v,lambda) THEN Eigenvector(A^k,v,lambda^k) for integer k>=0.
R09 IF Eigenvector(A,v,lambda) AND polynomial p THEN Eigenvector(p(A),v,p(lambda)).
R10 IF Eigenvector(A,v,lambda) AND A invertible THEN lambda!=0.
R11 IF Eigenvector(A,v,lambda) AND A invertible THEN Eigenvector(A^(-1),v,lambda^(-1)).
R12 IF Eigenvector(A,v,lambda) AND mu!=lambda THEN v not in E_mu.
R13 IF lambda_i pairwise distinct AND v_i corresponding eigenvectors THEN {v_i} linearly independent.
R14 IF n independent eigenvectors exist in F^n THEN A diagonalizable over F.
R15 IF A diagonalizable over F THEN an eigenbasis exists over F.
R16 IF A real symmetric THEN eigenvectors for distinct eigenvalues are orthogonal.
R17 IF A complex Hermitian THEN eigenvectors for distinct eigenvalues are orthogonal.
R18 IF A real symmetric THEN eigenvalues are real.
R19 IF A complex Hermitian THEN eigenvalues are real.
R20 IF A=P*D*P^(-1) AND D diagonal AND P invertible THEN columns(P) are eigenvectors of A associated with diagonal entries(D).
R21 IF B=P^(-1)*A*P AND A*v=lambda*v THEN B*(P^(-1)*v)=lambda*(P^(-1)*v).
R22 IF A*v=lambda*v AND B*v=mu*v THEN (A+B)*v=(lambda+mu)*v.
R23 IF A*v=lambda*v AND B*v=mu*v THEN (A*B)*v=lambda*mu*v.
R24 IF A*v=lambda*v THEN (c*A)*v=(c*lambda)*v.
R25 IF A*v=0 AND v!=0 THEN zero is eigenvalue of A.
R26 IF zero is eigenvalue of square A THEN A is singular.
R27 IF A is invertible THEN zero is not an eigenvalue.
R28 IF E_lambda has basis {b_1,...,b_k} THEN each b_i is eigenvector (basis members nonzero).
R29 IF v,w in E_lambda THEN v+w in E_lambda, possibly zero.
R30 IF v in E_lambda AND c in F THEN c*v in E_lambda, possibly zero.
R31 IF eigenvalue lambda has algebraic multiplicity one THEN its eigenspace has dimension one.
R32 IF sum of eigenspace dimensions for distinct eigenvalues equals n THEN A is diagonalizable over F.
R33 IF A is triangular THEN diagonal entries are eigenvalues (counting multiplicity) over F.
R34 IF A*v=lambda*v AND v!=0 THEN lambda is a root of characteristic polynomial.
R35 IF A*v=lambda*v THEN A^0*v=v.
R36 IF A has repeated eigenvalues THEN diagonalizability cannot be inferred without eigenspace dimensions.

08. VALIDATION CONSTRAINTS
V01: Reject zero vector as eigenvector.
V02: Reject non-square matrix for standard eigenpair operation.
V03: Reject incompatible vector/matrix dimensions.
V04: Verify field membership of entries, lambda and vector.
V05: Distinguish exact proof from floating-point tolerance checks.
V06: Do not infer orthogonality without normality/self-adjoint hypotheses as applicable.
V07: Do not infer real eigenvectors for all real matrices.
V08: Distinguish algebraic and geometric multiplicity.
V09: Verify P invertible before similarity transformations.
V10: Do not equate zero vector with an eigenvector merely because it lies in every eigenspace.
V11: Treat numerical eigenpairs as approximations, not symbolic proofs.
V12: Preserve eigenvalue field and basis convention in all transformations.

09. MATHEMATICAL EXAMPLES
EX01: A=diag(2,3); e1=(1,0) has lambda=2; e2=(0,1) has lambda=3.
EX02: A=I_n; every nonzero vector is eigenvector with lambda=1.
EX03: A=0_n; every nonzero vector is eigenvector with lambda=0.
EX04: A=diag(4,4); any nonzero vector in F^2 is eigenvector with lambda=4.
EX05: A=[[2,1],[0,2]]; (1,0) is eigenvector lambda=2; A not diagonalizable.
EX06: A=[[0,-1],[1,0]] over R; no real eigenvector; over C eigenvalues i,-i.
EX07: A=[[1,0],[0,-1]]; (1,0) lambda=1; (0,1) lambda=-1.
EX08: A=[[2,0],[0,3]]; (2,0) eigenvector lambda=2, illustrating nonzero scaling.
EX09: A=[[1,1],[1,1]]; (1,1) lambda=2 and (1,-1) lambda=0.
EX10: A=[[3,0],[0,5]]; A^2 has same coordinate eigenvectors, eigenvalues 9 and 25.
EX11: A=[[2,0],[0,4]]; A^(-1) has eigenvalues 1/2 and 1/4.
EX12: A=[[2,0],[0,2]]; E_2=F^2 includes zero, but zero is not eigenvector.
EX13: A=[[1,2],[2,1]]; (1,1) lambda=3 and (1,-1) lambda=-1, orthogonal.
EX14: A=[[0,1],[1,0]]; (1,1) lambda=1 and (1,-1) lambda=-1.
EX15: A=diag(1,2,3); three standard basis eigenvectors span F^3.
EX16: A=[[1,1],[0,1]]; characteristic polynomial (t-1)^2 but eigenspace dimension 1.

10. SEMANTIC_EQUIVALENCE
Eigenvector ~= characteristic_direction [context: linear operator; avoid implying Euclidean direction in arbitrary field]
Eigenvector ~= proper_vector [historical synonym]
Eigenvalue ~= characteristic_root [historical synonym]
Eigenspace == nullspace(A-lambda*I)
Eigenpair == ordered_pair(eigenvalue,eigenvector) with nonzero-vector constraint
NOT_EQUIVALENT(Eigenvector, ZeroVector)
NOT_EQUIVALENT(Eigenvector, ArbitraryVector)
NOT_EQUIVALENT(Eigenspace, SetOfEigenvectorsOnly)

11. KNOWLEDGE GRAPH TRIPLES
(Eigenvector, rdf:type, LinearAlgebraEntity)
(Eigenvector, requires, NonzeroVector)
(Eigenvector, associatedWith, Eigenvalue)
(Eigenvector, satisfies, EigenEquation)
(EigenEquation, hasForm, A*v=lambda*v)
(Eigenspace, equals, KernelOfShiftedOperator)
(Eigenvalue, rootOf, CharacteristicPolynomial)
(Diagonalisation, requires, Eigenbasis)
(SymmetricMatrix, admits, OrthonormalEigenbasisOverRealField)
(HermitianMatrix, admits, OrthonormalEigenbasisOverComplexField)

12. CONFIDENCE
confidence.model: 0.99 [standard mathematical statements with explicit hypotheses]
confidence.example_checks: exact_symbolic
confidence.source_alignment: unverified
confidence.policy: distinguish theorem certainty from numerical uncertainty and source provenance

13. FLOW_HASH
flow_hash.algorithm: SHA-256
flow_hash.canonicalisation: UTF-8; normalise line endings to LF; hash canonical payload excluding mutable hash-value field
flow_hash.value: COMPUTE_ON_PUBLICATION
flow_hash.purpose: reproducibility, integrity and version comparison

14. UPDATE_MODEL
update.version: 4.0
update.date: 2026-10-13
update.strategy: semantic versioning with theorem-level regression tests
update.triggers: changed axioms, added relation types, proof corrections, changed source-page content
update.audit: preserve prior hashes, author attribution, changed rules and verification results

15. AI_CONSUMPTION
ai_consumption.mode: typed_ontology_plus_rule_based_reasoning
ai_consumption.primary_entity: Eigenvector
ai_consumption.inputs: matrix_or_operator, scalar_field, candidate_vector, candidate_eigenvalue
ai_consumption.outputs: eigenpair_verification, eigenspace_basis, spectral_relations, proof_trace, confidence
ai_consumption.requirements: enforce nonzero constraint, dimensions, field, theorem hypotheses
ai_consumption.failure_states: INVALID_DIMENSIONS, ZERO_VECTOR, FIELD_MISMATCH, NON_EIGENPAIR, NUMERICAL_UNCERTAINTY
ai_consumption.provenance: include ontology ID, version, source URL and flow hash when published
ai_consumption.ethics: openly disclose AI co-authorship and mathematical verification status

END FLOW ALGEBRA