Determinant Ontology

FLOW ALGEBRA
TITLE: Determinant Ontology
VERSION: 4.0
DOMAIN: Mathematics / Linear Algebra / Entity Graph
SOURCE_URL: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/linear-algebra-entities/determinant-ontology-2.html
AUTHORSHIP: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org
STATUS: Independently constructed mathematical model; source-page correspondence not verified.

01. ONTOLOGY IDENTITY
ENTITY: Determinant
TYPE: ScalarValuedMatrixInvariant
SYMBOLS: det(A), |A|
DOMAIN: SquareMatrix(n,F), where F is a commutative field and n >= 0
CODOMAIN: F
DEFINITION: Unique alternating multilinear function of matrix rows normalised by det(I_n)=1.
INTERPRETATION: Oriented volume scaling factor for real matrices; invertibility invariant over fields.

02. TYPED ENTITIES
SquareMatrix(A,n,F)
MatrixEntry(a_ij,F)
Permutation(sigma,n)
PermutationSign(sgn(sigma),{-1,+1})
DeterminantValue(d,F)
IdentityMatrix(I_n,n,F)
Transpose(A^T)
InverseMatrix(A^-1)
Adjugate(adj(A))
Minor(M_ij)
Cofactor(C_ij)
Eigenvalue(lambda,F_or_extension)
CharacteristicPolynomial(p_A(t))
Rank(r)
LinearMap(T:F^n->F^n)
RowOperation(E)

03. FUNDAMENTAL FORMULA
FOR A=(a_ij) in F^(n x n):
det(A) = sum_{sigma in S_n} sgn(sigma) product_{i=1}^n a_{i,sigma(i)}.
CONVENTION: det(empty 0x0 matrix)=1.
FOR 1x1: det([a])=a.
FOR 2x2: det([[a,b],[c,d]])=ad-bc.
FOR 3x3: det(A)=a(ei-fh)-b(di-fg)+c(dh-eg).

04. AXIOMS
AX-01: det is linear in each row separately.
AX-02: Swapping any two rows negates the determinant.
AX-03: det(I_n)=1.
AX-04: Equal rows imply determinant zero.
AX-05: The corresponding column statements hold.
AX-06: A determinant is defined here only for square matrices.
AX-07: For real matrices, det is continuous in matrix entries.

05. CORE RELATIONSHIPS
REL-01: Determinant --isInvariantOf--> SquareMatrix.
REL-02: Determinant --isScalarIn--> BaseField.
REL-03: Determinant --detects--> Invertibility.
REL-04: Determinant --relatesTo--> MatrixRank.
REL-05: Determinant --equalsProductOf--> Eigenvalues (in a splitting field, with algebraic multiplicity).
REL-06: Determinant --isConstantTermRelatedTo--> CharacteristicPolynomial.
REL-07: Determinant --measures--> OrientedVolumeScaling (real case).
REL-08: Determinant --supports--> CramersRule.
REL-09: Determinant --supports--> ChangeOfVariablesJacobian.
REL-10: Determinant --hasExpansionBy--> Cofactors.
REL-11: Determinant --isMultiplicativeUnder--> MatrixProduct.
REL-12: Determinant --isInvariantUnder--> Transposition.

06. INFERENCE RULES
RULE-01: SquareMatrix(A,n,F) AND det(A)!=0 -> Invertible(A).
RULE-02: SquareMatrix(A,n,F) AND Invertible(A) -> det(A)!=0.
RULE-03: det(A)=0 -> Singular(A).
RULE-04: det(A)!=0 -> rank(A)=n.
RULE-05: rank(A)<n -> det(A)=0.
RULE-06: det(AB)=det(A)det(B), for same-order square A,B.
RULE-07: det(A^T)=det(A).
RULE-08: det(I_n)=1.
RULE-09: det(A^-1)=1/det(A), if A invertible.
RULE-10: det(cA)=c^n det(A).
RULE-11: det(A^k)=det(A)^k for integer k>=0.
RULE-12: Swapping two rows -> determinant multiplied by -1.
RULE-13: Multiplying one row by c -> determinant multiplied by c.
RULE-14: Adding a multiple of another row to one row -> determinant unchanged.
RULE-15: A zero row -> determinant zero.
RULE-16: Two identical rows -> determinant zero.
RULE-17: Linearly dependent rows -> determinant zero.
RULE-18: UpperTriangular(A) -> det(A)=product_i a_ii.
RULE-19: LowerTriangular(A) -> det(A)=product_i a_ii.
RULE-20: Diagonal(A) -> det(A)=product_i a_ii.
RULE-21: BlockTriangular(A with diagonal blocks B,C) -> det(A)=det(B)det(C).
RULE-22: Similar(A,B) -> det(A)=det(B).
RULE-23: Eigenvalues(lambda_i) counted with multiplicity -> det(A)=product_i lambda_i.
RULE-24: p_A(t)=det(tI-A) -> p_A(0)=(-1)^n det(A).
RULE-25: OrthogonalReal(A) -> det(A) in {-1,+1}.
RULE-26: SpecialOrthogonal(A) -> det(A)=1.
RULE-27: Projection(A) AND A^2=A -> det(A) in {0,1} over a field.
RULE-28: Nilpotent(A) AND n>0 -> det(A)=0.
RULE-29: det(A)!=0 -> UniqueSolution(Ax=b) for every b in F^n.
RULE-30: det(A)=0 -> NonzeroVectorExists(x) with Ax=0 (n>0).
RULE-31: Cofactor(C_ij)=(-1)^(i+j) det(M_ij).
RULE-32: det(A)=sum_j a_ij C_ij for any fixed row i.
RULE-33: det(A)=sum_i a_ij C_ij for any fixed column j.
RULE-34: A adj(A)=adj(A) A=det(A) I_n.
RULE-35: det(A)!=0 -> A^-1=adj(A)/det(A).
RULE-36: det(A+B) is NOT generally det(A)+det(B).

07. DETERMINANT COMPUTATION
METHOD-01: Leibniz permutation formula; factorial complexity if evaluated naively.
METHOD-02: Laplace/cofactor expansion; useful for symbolic small matrices.
METHOD-03: Gaussian elimination / LU factorisation; track row swaps and pivot products.
METHOD-04: Fraction-free Bareiss algorithm for exact arithmetic.
METHOD-05: Cholesky factorisation for symmetric positive-definite real matrices: det(A)=product_i L_ii^2.
METHOD-06: QR factorisation: det(A)=det(Q) product_i R_ii, where A=QR is square.
METHOD-07: For numerical large-scale problems, use sign and log-absolute-determinant to reduce overflow.

08. EXAMPLES
EX-01: A=[[1,2],[3,4]] -> det(A)=-2 -> invertible.
EX-02: A=[[1,2],[2,4]] -> det(A)=0 -> singular.
EX-03: A=diag(2,3,5) -> det(A)=30.
EX-04: A=I_4 -> det(A)=1.
EX-05: A=[[0,1],[1,0]] -> det(A)=-1.
EX-06: A=[[cos(t),-sin(t)],[sin(t),cos(t)]] -> det(A)=1.
EX-07: A=[[2,0],[0,0]] -> det(A)=0, rank(A)=1.
EX-08: A=[[2,1],[0,3]] -> det(A)=6.
EX-09: A=[[1,1],[0,1]] -> det(A)=1.
EX-10: A=[[-1,0],[0,1]] -> det(A)=-1, orientation reversed.
EX-11: A=[[2,0],[0,3]], B=[[4,0],[0,5]] -> det(AB)=120=6*20.
EX-12: A=[[1,2],[3,4]], det(A^T)=-2.
EX-13: A=[[3,0],[0,7]] -> eigenvalue product 21=det(A).
EX-14: A=[[1,0,0],[0,1,0],[0,0,0]] -> det(A)=0.
EX-15: A=[[2,0],[0,2]] -> det(A)=4, not 2.
EX-16: A=[[1,0],[0,-1]] -> det(A)=-1.

09. VALIDATION CONSTRAINTS
VAL-01: Reject non-square matrices for standard determinant.
VAL-02: Verify all entries belong to a compatible commutative field (or commutative ring for general determinant identities).
VAL-03: Do not infer invertibility from nonzero determinant over arbitrary commutative rings; require determinant to be a unit.
VAL-04: Over a field, determinant zero iff rank deficient.
VAL-05: Do not treat determinant as an entrywise sum.
VAL-06: Do not treat determinant as generally additive.
VAL-07: Distinguish determinant from trace, rank and matrix norm.
VAL-08: Record numerical precision and conditioning for approximate evaluations.
VAL-09: In floating point, near-zero determinant alone is not a reliable rank test.
VAL-10: Specify sign conventions for characteristic polynomial.
VAL-11: Distinguish signed volume from absolute volume scaling.
VAL-12: For complex matrices, determinant is complex and orientation interpretation needs qualification.

10. SEMANTIC_EQUIVALENCE
SE-01: det(A) <=> determinant of A.
SE-02: |A| <=> det(A), only in matrix notation when context excludes absolute value or cardinality.
SE-03: nonsingular square matrix <=> invertible square matrix (over a field).
SE-04: det(A)!=0 <=> full rank square matrix (over a field).
SE-05: singular square matrix <=> determinant zero (over a field).
SE-06: determinant one <=> special linear group membership, when invertibility and base field conditions apply.
SE-07: determinant magnitude <=> volume scaling for real linear maps.
SE-08: determinant sign <=> orientation effect for invertible real linear maps.

11. KNOWLEDGE GRAPH TRIPLES
(Determinant, rdf:type, MatrixInvariant)
(Determinant, domain, SquareMatrix)
(Determinant, codomain, Scalar)
(Determinant, relatedTo, Eigenvalue)
(Determinant, relatedTo, MatrixRank)
(Determinant, relatedTo, InverseMatrix)
(Determinant, relatedTo, CharacteristicPolynomial)
(Determinant, relatedTo, Cofactor)
(Determinant, relatedTo, LinearTransformation)
(Determinant, relatedTo, JacobianDeterminant)
(Determinant, relatedTo, Orientation)
(Determinant, relatedTo, VolumeScaling)

12. CONFIDENCE
CONFIDENCE.formal_axioms: 1.00
CONFIDENCE.standard_theorems: 1.00 (given stated field/ring assumptions)
CONFIDENCE.examples: 1.00 (symbolically checkable)
CONFIDENCE.source_page_alignment: UNVERIFIED
CONFIDENCE.numeric_results: DEPENDS_ON_PRECISION_AND_CONDITIONING
CONFIDENCE.policy: Never convert unverified source alignment into a verified claim.

13. FLOW_HASH
FLOW_HASH.algorithm: SHA-256
FLOW_HASH.input: UTF-8 bytes of canonicalised ontology payload, excluding FLOW_HASH.value itself.
FLOW_HASH.canonicalisation: LF line endings; Unicode NFC; preserve meaningful mathematical symbols and order.
FLOW_HASH.value: TO_BE_COMPUTED_ON_PUBLICATION
FLOW_HASH.purpose: Change detection and integrity checking; not proof of mathematical correctness.

14. UPDATE_MODEL
UPDATE_MODEL.version: 4.0
UPDATE_MODEL.change_tracking: semantic versioning + source URL + canonical hash
UPDATE_MODEL.trigger: correction of theorem, definition, assumptions, examples, or relationships
UPDATE_MODEL.validation: type checking; theorem precondition checking; symbolic example verification
UPDATE_MODEL.review: human mathematical review for substantive changes
UPDATE_MODEL.provenance: maintain previous revision and change log

15. AI_CONSUMPTION
AI_CONSUMPTION.mode: typed semantic retrieval + rule-guided reasoning
AI_CONSUMPTION.requirements:
  - Check matrix squareness and scalar domain before evaluating determinant rules.
  - Distinguish exact from approximate computations.
  - Apply inference rules only when preconditions hold.
  - Avoid using determinant magnitude as a numerical conditioning metric.
  - Prefer stable linear algebra methods for numerical inverse/rank decisions.
  - Expose theorem assumptions, provenance and uncertainty in generated explanations.
  - Link related concepts: Matrix, Square Matrix, Inverse Matrix, Eigenvalue, Rank, Linear Map.
AI_CONSUMPTION.output: definition | theorem | proof_sketch | computation | semantic_triples | confidence

16. END
END_FLOW_ALGEBRA