Eigenvalue Ontology

FLOW Eigenvalue_Ontology_v1.0

PAGE:
  title: "Eigenvalue Ontology"
  canonical_url: "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/eigenvalue-ontology.html"
  language: "en-GB"
  domain: "Mathematics > Algebra > Linear Algebra > Eigenvalues"

DEFINE EigenvalueOntology := {

  CORE_IDENTITY: {
    entity_type: "Mathematical Ontology",
    primary_entity: Eigenvalue,
    parent_entities: [LinearAlgebraConcept, SpectralConcept],
    related_entities: [
      LinearTransformation,
      Matrix,
      Vector,
      Eigenvector,
      Eigenspace,
      CharacteristicPolynomial,
      CharacteristicEquation,
      Determinant,
      Trace,
      AlgebraicMultiplicity,
      GeometricMultiplicity,
      Diagonalisation,
      Spectrum
    ]
  },

  DEFINITION: {
    meaning:
      "An eigenvalue of a linear transformation is a scalar lambda for which there exists a nonzero vector v satisfying T(v)=lambda*v.",
    canonical_relation:
      T(v)=lambda*v,
    constraints: [
      v != 0,
      lambda belongs_to_scalar_field
    ]
  },

  MATRIX_FORM: {
    relation:
      A*v=lambda*v,
    equivalent:
      (A-lambda*I)*v=0
  },

  CHARACTERISTIC_EQUATION: {
    equation:
      det(A-lambda*I)=0,
    purpose:
      determine_eigenvalues_of_square_matrix
  },

  CHARACTERISTIC_POLYNOMIAL: {
    notation:
      p_A(lambda)=det(A-lambda*I),
    roots:
      eigenvalues_of_A
  },

  SEMANTIC_PRIMITIVES: {
    Eigenvalue:
      scalar_scaling_factor_preserved_under_linear_action,
    Eigenvector:
      nonzero_vector_whose_direction_is_preserved,
    Eigenspace:
      kernel_of_(A-lambda*I),
    Spectrum:
      set_or_multiset_of_eigenvalues,
    AlgebraicMultiplicity:
      root_multiplicity_in_characteristic_polynomial,
    GeometricMultiplicity:
      dimension_of_eigenspace
  },

  EIGENVECTOR_RELATION: {
    lambda_is_eigenvalue:
      EXISTS v!=0:
        A*v=lambda*v,

    corresponding_eigenvectors:
      nonzero_elements_of_ker(A-lambda*I)
  },

  EIGENSPACE: {
    definition:
      E_lambda=ker(A-lambda*I),

    includes:
      zero_vector,

    eigenvectors:
      E_lambda\{0}
  },

  SPECTRUM: {
    notation:
      sigma(A),

    finite_dimensional_matrix_case:
      set_of_roots_of_characteristic_polynomial_over_algebraic_closure
  },

  ALGEBRAIC_MULTIPLICITY: {
    definition:
      multiplicity_of_lambda_as_root_of_characteristic_polynomial
  },

  GEOMETRIC_MULTIPLICITY: {
    definition:
      dim(E_lambda),

    relation:
      1 <= geometric_multiplicity <= algebraic_multiplicity
      for_eigenvalue_in_finite_dimensional_case
  },

  TRACE_RELATION: {
    over_algebraic_closure_counting_multiplicity:
      trace(A)=sum_of_eigenvalues
  },

  DETERMINANT_RELATION: {
    over_algebraic_closure_counting_multiplicity:
      det(A)=product_of_eigenvalues
  },

  ZERO_EIGENVALUE: {
    lambda=0_is_eigenvalue
      IFF A_is_singular,

    equivalent:
      det(A)=0
  },

  INVERTIBILITY_RELATION: {
    A_invertible
      IFF 0_not_in_spectrum
  },

  DIAGONALISATION: {
    A_diagonalisable:
      exists_basis_of_eigenvectors,

    matrix_relation:
      A=P*D*P^(-1),

    D:
      diagonal_matrix_of_eigenvalues
  },

  DISTINCT_EIGENVALUES: {
    theorem:
      eigenvectors_corresponding_to_distinct_eigenvalues_are_linearly_independent
  },

  DIAGONALISABILITY_SUFFICIENT_CONDITION: {
    n_by_n_matrix_with_n_distinct_eigenvalues:
      diagonalisable=TRUE
  },

  DIAGONALISABILITY_MULTIPLICITY_CONDITION: {
    finite_dimensional:
      A_diagonalisable
      IFF
      sum_of_geometric_multiplicities=n
      under_split_characteristic_polynomial_context
  },

  MINIMAL_POLYNOMIAL_RELATION: {
    eigenvalues:
      roots_of_minimal_polynomial,

    diagonalisation_condition:
      minimal_polynomial_splits_into_distinct_linear_factors
  },

  TRIANGULAR_MATRIX_RELATION: {
    eigenvalues:
      diagonal_entries
      counting_algebraic_multiplicity
  },

  SIMILARITY_INVARIANCE: {
    IF B=P^(-1)*A*P:
      characteristic_polynomial_same,
      eigenvalues_same,
      trace_same,
      determinant_same
  },

  SCALAR_MATRIX: {
    A=c*I:

    spectrum:
      {c},

    eigenspace:
      entire_vector_space
  },

  IDENTITY_MATRIX: {
    eigenvalue:
      1,

    eigenspace:
      entire_vector_space
  },

  NILPOTENT_MATRIX: {
    A^k=0:

    only_eigenvalue:
      0
  },

  IDEMPOTENT_MATRIX: {
    A^2=A:

    possible_eigenvalues:
      [0,1]
  },

  INVOLUTION_MATRIX: {
    A^2=I:

    possible_eigenvalues:
      [1,-1]
  },

  ORTHOGONAL_MATRIX_RELATION: {
    real_orthogonal_matrix:
      complex_eigenvalues_have_modulus_1
  },

  UNITARY_MATRIX_RELATION: {
    eigenvalues:
      modulus_1
  },

  HERMITIAN_MATRIX_RELATION: {
    eigenvalues:
      real,

    eigenvectors_for_distinct_eigenvalues:
      orthogonal
  },

  REAL_SYMMETRIC_MATRIX_RELATION: {
    eigenvalues:
      real,

    orthogonal_diagonalisation:
      TRUE
  },

  POSITIVE_DEFINITE_RELATION: {
    symmetric_or_Hermitian_matrix:
      positive_definite
      IFF all_eigenvalues_positive
  },

  POSITIVE_SEMIDEFINITE_RELATION: {
    eigenvalues:
      nonnegative
  },

  MATRIX_POWER_RELATION: {
    IF A*v=lambda*v:
      A^k*v=lambda^k*v
      for_integer_k>=0
  },

  POLYNOMIAL_FUNCTION_OF_MATRIX: {
    IF A*v=lambda*v:
      p(A)*v=p(lambda)*v
  },

  INVERSE_MATRIX_RELATION: {
    IF A*v=lambda*v
    AND lambda!=0:
      A^(-1)*v=(1/lambda)*v
  },

  SHIFT_RELATION: {
    IF A*v=lambda*v:
      (A+c*I)*v=(lambda+c)*v
  },

  SCALE_RELATION: {
    IF A*v=lambda*v:
      (c*A)*v=(c*lambda)*v
  },

  SPECTRAL_RADIUS: {
    definition:
      rho(A)=max{|lambda|: lambda in sigma(A)}
  },

  DYNAMICAL_SYSTEM_RELATION: {
    iteration:
      x_(k+1)=A*x_k,

    eigenvalue_magnitude:
      controls_growth_or_decay_along_eigendirections
  },

  DIFFERENTIAL_EQUATION_RELATION: {
    system:
      x'(t)=A*x(t),

    eigenpairs:
      produce_solutions_of_form
      e^(lambda*t)*v
  },

  STABILITY_RELATION: {
    discrete_system:
      eigenvalues_inside_unit_circle
      imply_asymptotic_decay_under_standard_conditions,

    continuous_linear_system:
      eigenvalues_with_negative_real_parts
      imply_asymptotic_decay_under_standard_conditions
  },

  MARKOV_CHAIN_RELATION: {
    stochastic_matrix:
      eigenvalue_1_common,

    stationary_distribution:
      related_to_eigenvector_for_eigenvalue_1
      depending_on_row_or_column_convention
  },

  GRAPH_THEORY_RELATION: {
    adjacency_or_laplacian_matrix:
      eigenvalues_encode_structural_graph_information
  },

  PCA_RELATION: {
    covariance_matrix:
      eigenvectors_define_principal_directions,

    eigenvalues:
      variances_along_principal_components
  },

  QUANTUM_MECHANICS_RELATION: {
    observable_operator:
      eigenvalues_correspond_to_possible_measurement_values
      in_standard_formalism
  },

  SYMBOLIC_RECOGNITION: {
    ACCEPT: [
      "Av=lambda v",
      "det(A-lambda I)=0",
      "lambda=3 is an eigenvalue",
      "spectrum of matrix A"
    ],

    REDIRECT_NONLINEAR_EIGENPROBLEM:
      nonlinear_operator_or_parameter_problem
      -> specialised_ontology
  },

  NATURAL_LANGUAGE_MAPPING: {
    "eigenvalue":
      scalar_preserved_by_linear_action_on_nonzero_vector,

    "characteristic value":
      historical_synonym_for_eigenvalue,

    "spectrum":
      collection_of_eigenvalues,

    "principal eigenvalue":
      context_dependent_distinguished_eigenvalue
  },

  ERROR_DETECTION: {
    ZeroVectorError:
      v=0_used_as_eigenvector
      -> reject,

    NonSquareMatrixError:
      ordinary_characteristic_polynomial_requested_for_nonsquare_matrix
      -> redirect_to_singular_value_or_generalised_context,

    MissingDeterminantEquation:
      solving_det(A-lambdaI)!=0_for_eigenvalues
      -> flag,

    MultiplicityConfusion:
      algebraic_and_geometric_multiplicity_conflated
      -> flag,

    DiagonalisationError:
      insufficient_independent_eigenvectors
      -> reject_diagonalisation_claim
  },

  ONTOLOGY_RELATIONS: {
    IS_A: [
      "Eigenvalue IS_A SpectralConcept"
    ],

    RELATED_TO: [
      "Eigenvalue RELATED_TO Eigenvector",
      "Eigenvalue RELATED_TO Eigenspace",
      "Eigenvalue RELATED_TO CharacteristicPolynomial",
      "Eigenvalue RELATED_TO Diagonalisation",
      "Eigenvalue RELATED_TO Spectrum",
      "Eigenvalue RELATED_TO LinearTransformation"
    ],

    DEPENDS_ON: [
      VectorSpace,
      LinearTransformation,
      ScalarField,
      Determinant
    ]
  },

  SEMANTIC_EQUIVALENCE: {
    equivalent_conditions: [
      "lambda eigenvalue of A",
      "exists nonzero v with Av=lambda v",
      "ker(A-lambda I) nontrivial",
      "det(A-lambda I)=0 in finite-dimensional square-matrix case"
    ]
  },

  CONFIDENCE: {
    eigenvalue_confidence:
      IF nonzero_eigenvector_verified
      THEN 1.0,

    characteristic_root_confidence:
      IF determinant_polynomial_computation_verified
      THEN 1.0,

    multiplicity_confidence:
      IF polynomial_root_and_eigenspace_dimensions_verified
      THEN 1.0
  },

  AI_CONSUMPTION: {
    tasks: [
      recognise_eigenvalue_problem,
      build_characteristic_polynomial,
      solve_characteristic_equation,
      verify_candidate_eigenvalues,
      compute_eigenspaces,
      distinguish_algebraic_and_geometric_multiplicity,
      test_diagonalisability,
      infer_trace_and_determinant_relations,
      analyse matrix powers,
      analyse dynamical stability,
      connect spectral information to applications,
      detect eigenvalue reasoning errors,
      link related ontologies
    ],

    preferred_representation:
      typed_spectral_relation_graph,

    reasoning_mode:
      recursive_characteristic_kernel_and_multiplicity_analysis
  },

  RECURSIVE_REASONING: {

    FUNCTION AnalyseEigenvalueProblem(A,Field,Context):

      IF not IsSquareLinearOperator(A):
        RETURN RedirectToGeneralisedSpectralOntology

      p := CharacteristicPolynomial(A)

      candidates := SolvePolynomial(
        p(lambda)=0,
        FieldOrAlgebraicClosure(Field)
      )

      results := []

      FOR EACH lambda IN candidates:
        E := Kernel(A-lambda*I)

        IF Dimension(E)>0:
          alg_mult := RootMultiplicity(p,lambda)
          geom_mult := Dimension(E)

          results += {
            eigenvalue:lambda,
            algebraic_multiplicity:alg_mult,
            geometric_multiplicity:geom_mult,
            eigenspace:E
          }

      diagonalisable :=
        TestDiagonalisationFromEigenspaces(results,Dimension(A))

      RETURN {
        characteristic_polynomial:p,
        spectrum:results,
        diagonalisable,
        verified:TRUE
      }
  },

  RECURSIVE_COHERENCE: {
    invariant_1:
      eigenvector_nonzero=TRUE,

    invariant_2:
      eigenspace_equals_kernel_of_A_minus_lambda_I=TRUE,

    invariant_3:
      geometric_multiplicity_not_exceed_algebraic_multiplicity=TRUE,

    invariant_4:
      trace_and_determinant_match_eigenvalue_sums_and_products_when_applicable=TRUE,

    invariant_5:
      similarity_preserves_spectrum=TRUE
  },

  FLOW_HASH: {
    semantic_signature_inputs: [
      ontology_name,
      eigenvalue_definition,
      eigenvector_relation,
      characteristic_polynomial,
      eigenspace_semantics,
      multiplicity_rules,
      diagonalisation_relations
    ],

    purpose:
      "Detect semantic drift between revisions of the ontology."
  },

  UPDATE_MODEL: {
    version: "1.0",

    immutable_constraints: [
      nonzero_eigenvector_requirement,
      Av_equals_lambda_v,
      eigenspace_kernel_relation,
      characteristic_equation_for_square_finite_dimensional_case,
      geometric_multiplicity_leq_algebraic_multiplicity
    ],

    revision_triggers: [
      semantic_conflict,
      VectorSpaceOntology_update,
      MatrixOntology_update,
      LinearTransformationOntology_update,
      spectral_theory_extension,
      representation_standard_change
    ]
  },

  KNOWLEDGE_GRAPH: {
    nodes: [
      Eigenvalue,
      Eigenvector,
      Eigenspace,
      Matrix,
      LinearTransformation,
      CharacteristicPolynomial,
      Determinant,
      AlgebraicMultiplicity,
      GeometricMultiplicity,
      Spectrum,
      Diagonalisation
    ],

    edges: [
      Eigenvalue -> ASSOCIATED_WITH -> Eigenvector,
      Eigenvalue -> HAS_EIGENSPACE -> Eigenspace,
      Eigenspace -> EQUALS -> KernelOfShiftedOperator,
      Matrix -> HAS_CHARACTERISTIC_POLYNOMIAL -> CharacteristicPolynomial,
      CharacteristicPolynomial -> HAS_ROOT -> Eigenvalue,
      Eigenvalue -> HAS_ALGEBRAIC_MULTIPLICITY -> AlgebraicMultiplicity,
      Eigenvalue -> HAS_GEOMETRIC_MULTIPLICITY -> GeometricMultiplicity,
      LinearTransformation -> HAS_SPECTRUM -> Spectrum,
      Spectrum -> SUPPORTS_ANALYSIS_OF -> Diagonalisation
    ]
  },

  CROSS_ONTOLOGY_LINKS: {
    VectorSpaceOntology:
      ambient_space,

    LinearTransformationOntology:
      operator_semantics,

    MatrixOntology:
      finite_dimensional_representation,

    DeterminantOntology:
      characteristic_polynomial,

    PolynomialEquationOntology:
      eigenvalue_equation_roots,

    ComplexNumberOntology:
      nonreal_eigenvalues,

    DifferentialEquationOntology:
      linear_system_solution_modes,

    DynamicalSystemsOntology:
      stability_and_growth,

    StatisticsOntology:
      covariance_eigenanalysis
  },

  TERMINATION_CONDITION: {
    ontology_complete_when: [
      eigenvalue_defined,
      eigenvector_relation_defined,
      characteristic_polynomial_defined,
      eigenspace_defined,
      spectrum_defined,
      multiplicities_defined,
      diagonalisation_relations_defined,
      transformation_rules_defined,
      application_links_defined,
      error_detection_defined,
      cross_ontology_links_defined,
      recursive_coherence_verified
    ]
  }

}

EXECUTE Eigenvalue_Ontology_v1.0:

  INPUT:
    mathematical_entity_or_query

  PIPELINE:
    Parse
    -> IdentifyLinearOperatorOrMatrix
    -> VerifySquareFiniteDimensionalContextWhenRequired
    -> ConstructCharacteristicPolynomial
    -> SolveCharacteristicEquation
    -> VerifyCandidateEigenvalues
    -> ComputeEigenspaces
    -> DetermineAlgebraicMultiplicity
    -> DetermineGeometricMultiplicity
    -> AnalyseDiagonalisation
    -> DeriveTraceDeterminantAndSpectralRelations
    -> ApplyApplicationSpecificSpectralReasoning
    -> VerifySemanticConsistency
    -> LinkRelatedOntologies
    -> AssignConfidence
    -> EmitSemanticResult

  OUTPUT: {
    entity,
    classification,
    operator_or_matrix,
    characteristic_polynomial,
    characteristic_equation,
    eigenvalues,
    spectrum,
    eigenspaces,
    eigenvectors,
    algebraic_multiplicities,
    geometric_multiplicities,
    trace_relation,
    determinant_relation,
    diagonalisation_status,
    spectral_radius,
    application_relations,
    verification,
    ontology_relations,
    confidence,
    semantic_equivalence,
    flow_hash
  }

END FLOW Eigenvalue_Ontology_v1.0