Determinant Ontology

FLOW Determinant_Ontology_v1.0

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

DEFINE DeterminantOntology := {

  CORE_IDENTITY: {
    entity_type: "Mathematical Ontology",
    primary_entity: Determinant,
    parent_entities: [LinearAlgebraConcept, ScalarInvariant],
    related_entities: [
      Matrix,
      SquareMatrix,
      LinearTransformation,
      Eigenvalue,
      Invertibility,
      Rank,
      VolumeScaling,
      Orientation,
      Cofactor,
      Minor,
      Permutation,
      CharacteristicPolynomial
    ]
  },

  DEFINITION: {
    meaning:
      "The determinant is a scalar associated with a square matrix or finite-dimensional linear transformation, encoding invertibility, oriented volume scaling, and structural information.",
    notation:
      det(A),
    matrix_requirement:
      A_is_square
  },

  SEMANTIC_PRIMITIVES: {
    DeterminantValue:
      scalar_in_base_field,
    Minor:
      determinant_of_submatrix_obtained_by_deleting_row_and_column,
    Cofactor:
      signed_minor,
    PermutationSign:
      parity_factor_in_Leibniz_formula,
    Orientation:
      sign_or_orientation_effect_in_real_geometry,
    VolumeScaling:
      absolute_value_of_determinant
  },

  DIMENSION_ONE: {
    det([a]):
      a
  },

  DIMENSION_TWO: {
    matrix:
      [[a,b],[c,d]],

    determinant:
      a*d-b*c
  },

  DIMENSION_THREE: {
    matrix:
      [[a,b,c],[d,e,f],[g,h,i]],

    determinant:
      a*(e*i-f*h)-b*(d*i-f*g)+c*(d*h-e*g)
  },

  LEIBNIZ_FORMULA: {
    determinant:
      det(A)=SUM_over_permutations_sigma(
        sign(sigma)*PRODUCT_i(a_i,sigma(i))
      ),

    semantic_components: [
      permutation,
      permutation_sign,
      product_selection,
      summation
    ]
  },

  COFACTOR_EXPANSION: {
    along_row_i:
      det(A)=SUM_j(a_ij*C_ij),

    cofactor:
      C_ij=(-1)^(i+j)*M_ij,

    minor:
      M_ij=det(submatrix_delete_row_i_column_j)
  },

  EXPANSION_INVARIANCE: {
    rule:
      cofactor_expansion_along_any_row_or_column_gives_same_determinant
  },

  TRIANGULAR_MATRIX: {
    IF A_is_upper_or_lower_triangular:
      det(A)=PRODUCT(diagonal_entries)
  },

  DIAGONAL_MATRIX: {
    determinant:
      product_of_diagonal_entries
  },

  IDENTITY_MATRIX: {
    det(I_n):
      1
  },

  ZERO_MATRIX: {
    det(0_n):
      0
      for_n>=1
  },

  ROW_OPERATION_RULES: {

    RowSwap: {
      effect:
        determinant_multiplied_by_-1
    },

    RowScaling: {
      R_i -> c*R_i,
      effect:
        determinant_multiplied_by_c
    },

    RowReplacement: {
      R_i -> R_i+c*R_j,
      effect:
        determinant_unchanged
    }
  },

  COLUMN_OPERATION_RULES: {
    analogous_to_row_operation_rules:
      TRUE
  },

  MULTILINEARITY: {
    determinant:
      linear_in_each_row_separately
      when_other_rows_fixed,

    also:
      linear_in_each_column_separately
  },

  ALTERNATING_PROPERTY: {
    IF two_rows_equal:
      determinant=0,

    IF two_columns_equal:
      determinant=0
  },

  LINEAR_DEPENDENCE_RELATION: {
    IF rows_or_columns_linearly_dependent:
      determinant=0
  },

  INVERTIBILITY: {
    A_invertible
      IFF det(A)!=0
  },

  SINGULARITY: {
    A_singular
      IFF det(A)=0
  },

  INVERSE_FORMULA_RELATION: {
    A^(-1):
      adj(A)/det(A),

    condition:
      det(A)!=0
  },

  ADJUGATE_RELATION: {
    identity:
      A*adj(A)=adj(A)*A=det(A)*I
  },

  PRODUCT_RULE: {
    det(A*B):
      det(A)*det(B)
  },

  POWER_RULE: {
    det(A^k):
      det(A)^k
      for_integer_k>=0
  },

  INVERSE_RULE: {
    IF A_invertible:
      det(A^(-1))=1/det(A)
  },

  TRANSPOSE_RULE: {
    det(A^T):
      det(A)
  },

  SCALAR_MULTIPLICATION: {
    n_by_n_matrix_A:
      det(c*A)=c^n*det(A)
  },

  SIMILARITY_INVARIANCE: {
    IF B=P^(-1)*A*P:
      det(B)=det(A)
  },

  TRACE_CONTRAST: {
    determinant:
      multiplicative_invariant,

    trace:
      additive_linear_invariant,

    relation_to_eigenvalues:
      determinant_product_vs_trace_sum
  },

  EIGENVALUE_RELATION: {
    over_algebraic_closure_counting_multiplicity:
      det(A)=PRODUCT(eigenvalues)
  },

  CHARACTERISTIC_POLYNOMIAL: {
    common_convention:
      p_A(lambda)=det(lambda*I-A)
      OR
      det(A-lambda*I),

    warning:
      sign_convention_depends_on_definition,

    eigenvalue_condition:
      determinant_equals_zero
  },

  ZERO_EIGENVALUE: {
    0_is_eigenvalue
      IFF det(A)=0
  },

  RANK_RELATION: {
    n_by_n_matrix:
      det(A)!=0
      IFF rank(A)=n
  },

  LINEAR_SYSTEM_RELATION: {
    A*x=b,

    IF det(A)!=0:
      unique_solution,

    IF det(A)=0:
      no_unique_solution
      and_requires_consistency_analysis
  },

  CRAMERS_RULE: {
    condition:
      square_system
      AND det(A)!=0,

    solution:
      x_i=det(A_i)/det(A)
  },

  GEOMETRIC_MEANING: {
    linear_transformation_T:
      determinant_is_oriented_volume_scaling_factor
  },

  AREA_SCALING: {
    in_2D:
      |det(A)|=area_scale_factor
  },

  VOLUME_SCALING: {
    in_3D:
      |det(A)|=volume_scale_factor
  },

  ORIENTATION: {
    real_matrix:
      det(A)>0
      -> orientation_preserving,

    det(A)<0:
      orientation_reversing,

    det(A)=0:
      dimension_collapsing
  },

  PARALLELEPIPED_RELATION: {
    vectors_as_columns:
      absolute_determinant_equals_volume_of_parallelepiped
  },

  CROSS_PRODUCT_RELATION: {
    in_R3:
      scalar_triple_product
      equals_determinant_of_matrix_with_vectors_as_rows_or_columns
      under_consistent_orientation
  },

  CHANGE_OF_VARIABLES_RELATION: {
    multivariable_calculus:
      Jacobian_determinant_controls_local_volume_scaling
  },

  JACOBIAN_RELATION: {
    JacobianMatrix:
      derivative_matrix_of_multivariable_map,

    JacobianDeterminant:
      local_oriented_volume_scale
  },

  BLOCK_MATRIX_RELATION: {
    block_triangular_matrix:
      determinant_equals_product_of_determinants_of_diagonal_blocks
  },

  BLOCK_DIAGONAL_MATRIX: {
    determinant:
      product_of_block_determinants
  },

  PERMUTATION_MATRIX: {
    determinant:
      sign_of_permutation
  },

  ORTHOGONAL_MATRIX: {
    determinant:
      ±1
  },

  SPECIAL_ORTHOGONAL_MATRIX: {
    determinant:
      1
  },

  UNITARY_MATRIX: {
    determinant_modulus:
      1
  },

  HERMITIAN_MATRIX: {
    determinant:
      real
      because_eigenvalues_real
  },

  POSITIVE_DEFINITE_MATRIX: {
    determinant:
      positive
      for_finite_dimensional_real_symmetric_or_complex_Hermitian_case
  },

  GRAM_MATRIX_RELATION: {
    Gram_determinant:
      squared_volume_of_parallelotope_spanned_by_vectors
  },

  VANDERMONDE_DETERMINANT: {
    matrix:
      powers_of_distinct_scalars_x_i,

    determinant:
      PRODUCT_{i<j}(x_j-x_i)
  },

  WRONSKIAN_RELATION: {
    determinant_of_function_derivative_matrix:
      Wronskian,

    use:
      tests_linear_independence_under_appropriate_conditions
  },

  RESULTANT_RELATION: {
    determinant_of_Sylvester_matrix:
      resultant_of_polynomials,

    semantic_use:
      common_root_detection
  },

  DETERMINANT_AS_ALTERNATING_FORM: {
    determinant:
      unique_alternating_multilinear_form_on_rows_or_columns
      normalised_by_det(I)=1
  },

  EXTERIOR_ALGEBRA_RELATION: {
    top_exterior_power:
      linear_map_induced_by_A_acts_by_scalar_det(A)
  },

  FIELD_DEPENDENCE: {
    determinant_defined_over:
      commutative_field_or_commutative_ring_contexts,

    noncommutative_ring:
      ordinary_determinant_requires_special_care_or_alternative_notions
  },

  COMPUTATION_STRATEGIES: {
    small_matrices:
      direct_formula_or_cofactor_expansion,

    sparse_matrices:
      strategic_cofactor_expansion,

    large_numeric_matrices:
      LU_factorisation_preferred,

    triangularised_matrix:
      product_of_diagonal_entries_with_row_swap_and_scaling_tracking
  },

  LU_RELATION: {
    A=P*L*U,

    determinant:
      det(A)=det(P)*det(L)*det(U),

    common_unit_lower_triangular_L:
      det(L)=1
  },

  NUMERICAL_STABILITY: {
    warning:
      explicit_cofactor_expansion_is_computationally_expensive_for_large_n,

    preferred:
      elimination_or_factorisation_methods
  },

  SYMBOLIC_RECOGNITION: {
    ACCEPT: [
      "det(A)",
      "|A| in determinant context",
      "ad-bc",
      "det(lambda I-A)"
    ],

    AMBIGUITY_WARNING:
      vertical_bars_may_mean_absolute_value_norm_or_determinant
  },

  NATURAL_LANGUAGE_MAPPING: {
    "determinant":
      scalar_invariant_of_square_matrix,

    "singular":
      determinant_zero,

    "nonsingular":
      determinant_nonzero,

    "volume scale factor":
      absolute_determinant,

    "orientation preserving":
      positive_determinant_in_real_finite_dimensional_context
  },

  ERROR_DETECTION: {
    NonSquareMatrixError:
      ordinary_determinant_requested_for_rectangular_matrix
      -> reject_or_redirect,

    RowSwapSignError:
      determinant_sign_not_changed
      -> flag,

    RowScaleError:
      determinant_scaling_not_tracked
      -> flag,

    CofactorSignError:
      (-1)^(i+j)_misapplied
      -> flag,

    InvertibilityError:
      det(A)=0_but_inverse_claimed
      -> flag,

    ProductRuleError:
      det(A+B)=det(A)+det(B)_assumed_generally
      -> flag
  },

  ONTOLOGY_RELATIONS: {
    IS_A: [
      "Determinant IS_A ScalarInvariant"
    ],

    RELATED_TO: [
      "Determinant RELATED_TO Matrix",
      "Determinant RELATED_TO Invertibility",
      "Determinant RELATED_TO Eigenvalue",
      "Determinant RELATED_TO VolumeScaling",
      "Determinant RELATED_TO Orientation",
      "Determinant RELATED_TO CharacteristicPolynomial"
    ],

    DEPENDS_ON: [
      SquareMatrix,
      ScalarField,
      PermutationSign,
      Multilinearity
    ]
  },

  SEMANTIC_EQUIVALENCE: {
    equivalent_invertibility_conditions: [
      "det(A)!=0",
      "A invertible",
      "rank(A)=n",
      "0 not an eigenvalue"
    ],

    geometric_equivalence:
      absolute_det_equals_volume_scaling_factor
  },

  CONFIDENCE: {
    computation_confidence:
      IF matrix_square
      AND operation_rules_verified
      THEN 1.0,

    invertibility_confidence:
      IF determinant_nonzero_verified
      THEN 1.0,

    geometric_confidence:
      IF real_linear_map_context_and_dimension_resolved
      THEN 1.0
  },

  AI_CONSUMPTION: {
    tasks: [
      recognise_determinant_expression,
      verify_square_matrix_requirement,
      compute_2x2_and_3x3_determinants,
      compute_by_cofactor_expansion,
      compute_by_elimination_or_LU,
      track_row_operation_effects,
      test invertibility,
      derive rank_and_eigenvalue_relations,
      interpret geometric scaling,
      evaluate characteristic determinants,
      detect determinant reasoning errors,
      link related ontologies
    ],

    preferred_representation:
      typed_scalar_invariant_with_algebraic_and_geometric_relations,

    reasoning_mode:
      recursive_matrix_reduction_and_invariant_tracking
  },

  RECURSIVE_REASONING: {

    FUNCTION AnalyseDeterminant(A,Context):

      parsed := ParseMatrix(A)

      IF not IsSquare(parsed):
        RETURN NotApplicable("ordinary determinant requires square matrix")

      n := Size(parsed)

      IF n=1:
        value := parsed[1,1]

      ELSE IF n=2:
        value := a11*a22-a12*a21

      ELSE IF IsTriangular(parsed):
        value := ProductDiagonal(parsed)

      ELSE:
        strategy := SelectDeterminantStrategy(parsed,Context)

        IF strategy=Elimination:
          value := DeterminantByRowReduction(parsed)

        ELSE IF strategy=LU:
          value := DeterminantByLU(parsed)

        ELSE:
          value := CofactorExpansion(parsed)

      properties := {
        invertible:(value!=0),
        singular:(value=0),
        orientation:
          IF RealMatrix(parsed)
          THEN SignInterpretation(value)
          ELSE NotApplicable
      }

      verified := VerifyDeterminantProperties(parsed,value)

      RETURN {
        entity:Determinant,
        matrix:parsed,
        value,
        properties,
        verified
      }
  },

  RECURSIVE_COHERENCE: {
    invariant_1:
      determinant_only_for_square_matrix_in_standard_form=TRUE,

    invariant_2:
      row_operation_effects_tracked_correctly=TRUE,

    invariant_3:
      det_AB_equals_det_A_times_det_B=TRUE,

    invariant_4:
      det_nonzero_matches_invertibility=TRUE,

    invariant_5:
      eigenvalue_product_relation_consistent_when_applicable=TRUE
  },

  FLOW_HASH: {
    semantic_signature_inputs: [
      ontology_name,
      determinant_definition,
      computation_rules,
      row_operation_effects,
      invertibility_relation,
      eigenvalue_relation,
      geometric_scaling_semantics
    ],

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

  UPDATE_MODEL: {
    version: "1.0",

    immutable_constraints: [
      square_matrix_requirement,
      multilinearity,
      alternating_property,
      det_identity_equals_one,
      product_rule,
      invertibility_equivalence
    ],

    revision_triggers: [
      semantic_conflict,
      MatrixOntology_update,
      EigenvalueOntology_update,
      LinearTransformationOntology_update,
      computational_strategy_update,
      representation_standard_change
    ]
  },

  KNOWLEDGE_GRAPH: {
    nodes: [
      Determinant,
      Matrix,
      SquareMatrix,
      Minor,
      Cofactor,
      Permutation,
      Invertibility,
      Rank,
      Eigenvalue,
      VolumeScaling,
      Orientation,
      CharacteristicPolynomial
    ],

    edges: [
      Determinant -> APPLIES_TO -> SquareMatrix,
      Determinant -> COMPUTED_FROM -> Minor,
      Minor -> SIGNED_AS -> Cofactor,
      Determinant -> DEPENDS_ON -> Permutation,
      Determinant -> DETERMINES -> Invertibility,
      Determinant -> RELATED_TO -> Rank,
      Determinant -> EQUALS_PRODUCT_OF -> Eigenvalue,
      Determinant -> REPRESENTS -> VolumeScaling,
      Determinant -> REPRESENTS -> Orientation,
      CharacteristicPolynomial -> DEFINED_BY -> Determinant
    ]
  },

  CROSS_ONTOLOGY_LINKS: {
    MatrixOntology:
      direct_structural_context,

    VectorSpaceOntology:
      finite_dimensional_linear_structure,

    LinearTransformationOntology:
      basis_independent_operator_invariant,

    EigenvalueOntology:
      product_and_characteristic_relations,

    RankOntology:
      full_rank_equivalence,

    SystemOfEquationsOntology:
      uniqueness_and_Cramers_rule,

    GeometryOntology:
      area_volume_orientation,

    CalculusOntology:
      Jacobian_determinant
  },

  TERMINATION_CONDITION: {
    ontology_complete_when: [
      determinant_defined,
      small_matrix_formulas_defined,
      permutation_and_cofactor_models_defined,
      row_operation_rules_defined,
      product_transpose_inverse_rules_defined,
      invertibility_and_rank_relations_defined,
      eigenvalue_and_characteristic_relations_defined,
      geometric_semantics_defined,
      computational_strategies_defined,
      error_detection_defined,
      cross_ontology_links_defined,
      recursive_coherence_verified
    ]
  }

}

EXECUTE Determinant_Ontology_v1.0:

  INPUT:
    mathematical_entity_or_query

  PIPELINE:
    Parse
    -> IdentifySquareMatrixOrLinearOperator
    -> SelectDeterminantRepresentation
    -> ComputeDeterminant
    -> VerifyRowOperationAccounting
    -> DetermineInvertibility
    -> RelateToRankAndEigenvalues
    -> DeriveGeometricScalingAndOrientation
    -> EvaluateCharacteristicOrJacobianContextWhenRelevant
    -> VerifySemanticConsistency
    -> LinkRelatedOntologies
    -> AssignConfidence
    -> EmitSemanticResult

  OUTPUT: {
    entity,
    classification,
    matrix,
    determinant_value,
    computation_method,
    minors,
    cofactors,
    invertibility,
    singularity,
    rank_relation,
    eigenvalue_relation,
    trace_contrast,
    orientation,
    volume_scaling,
    characteristic_polynomial_relation,
    jacobian_relation,
    verification,
    ontology_relations,
    confidence,
    semantic_equivalence,
    flow_hash
  }

END FLOW Determinant_Ontology_v1.0