Jacobi matrix
E183956
A Jacobi matrix is a tridiagonal matrix, often symmetric, that arises in numerical analysis and mathematical physics, particularly in the study of orthogonal polynomials and eigenvalue problems.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Jacobi matrix canonical | 5 |
How this entity was disambiguated
This entity first appeared as the object of triple T1615222 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Jacobi matrix Context triple: [Carl Gustav Jacob Jacobi, notableWork, Jacobi matrix]
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A.
Jacobi bracket
The Jacobi bracket is a bilinear operation generalizing the Poisson bracket in differential geometry, central to the theory of Jacobi manifolds and Hamiltonian systems.
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B.
Jacobi
Jacobi is a German surname most famously associated with the 19th-century mathematician Carl Gustav Jacob Jacobi, known for his foundational work in elliptic functions and number theory.
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C.
Jacobi operator
The Jacobi operator is a linear differential operator central to the theory of elliptic functions and integrable systems, named after the mathematician Carl Gustav Jacob Jacobi.
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D.
Jacobi method
The Jacobi method is an iterative numerical algorithm used to solve systems of linear equations by repeatedly updating each variable using values from the previous iteration.
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E.
Jacobi integral
The Jacobi integral is a conserved quantity in celestial mechanics and dynamical systems that simplifies the analysis of motion in rotating reference frames, particularly in the restricted three-body problem.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Jacobi matrix Target entity description: A Jacobi matrix is a tridiagonal matrix, often symmetric, that arises in numerical analysis and mathematical physics, particularly in the study of orthogonal polynomials and eigenvalue problems.
-
A.
Jacobi bracket
The Jacobi bracket is a bilinear operation generalizing the Poisson bracket in differential geometry, central to the theory of Jacobi manifolds and Hamiltonian systems.
-
B.
Jacobi
Jacobi is a German surname most famously associated with the 19th-century mathematician Carl Gustav Jacob Jacobi, known for his foundational work in elliptic functions and number theory.
-
C.
Jacobi operator
The Jacobi operator is a linear differential operator central to the theory of elliptic functions and integrable systems, named after the mathematician Carl Gustav Jacob Jacobi.
-
D.
Jacobi method
The Jacobi method is an iterative numerical algorithm used to solve systems of linear equations by repeatedly updating each variable using values from the previous iteration.
-
E.
Jacobi integral
The Jacobi integral is a conserved quantity in celestial mechanics and dynamical systems that simplifies the analysis of motion in rotating reference frames, particularly in the restricted three-body problem.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
matrix concept
ⓘ
tridiagonal matrix ⓘ |
| appearsIn |
inverse spectral problems
ⓘ
orthogonal polynomial ensembles ⓘ random matrix theory ⓘ theory of continued fractions ⓘ |
| canBe |
finite-dimensional
ⓘ
infinite-dimensional ⓘ |
| eigenvaluesCorrespondTo | nodes of Gaussian quadrature ⓘ |
| eigenvectorsCorrespondTo | values of orthogonal polynomials ⓘ |
| encodes | recurrence coefficients of orthogonal polynomials ⓘ |
| field |
approximation theory
ⓘ
eigenvalue problems ⓘ linear algebra ⓘ mathematical physics ⓘ numerical analysis ⓘ orthogonal polynomials ⓘ spectral theory ⓘ |
| generalizationOf | discrete one-dimensional Schrödinger operator ⓘ |
| hasDiagonalEntries | recurrence coefficients a_n ⓘ |
| hasMainDiagonal | real sequence ⓘ |
| hasOffDiagonalEntries | recurrence coefficients b_n ⓘ |
| hasProperty |
banded
ⓘ
often symmetric ⓘ real entries ⓘ sparse ⓘ tridiagonal ⓘ |
| hasSize | n×n ⓘ |
| hasSpectrum | real when symmetric and real ⓘ |
| hasSubDiagonal | real sequence ⓘ |
| hasSuperDiagonal | real sequence ⓘ |
| hasSymmetry | symmetric when sub- and super-diagonals coincide ⓘ |
| isOperatorOn | ℓ²(ℕ) in infinite-dimensional case ⓘ |
| namedAfter | Carl Gustav Jacob Jacobi ⓘ |
| relatedTo | orthogonal polynomial sequence ⓘ |
| specialCaseOf | band matrix ⓘ |
| spectralMeasure | associated orthogonality measure ⓘ |
| usedIn |
Gaussian quadrature
ⓘ
Golub–Kahan bidiagonalization ⓘ
linked to:
Bidiagonal
Jacobi eigenvalue algorithm ⓘ Lanczos algorithm ⓘ continued fraction representations ⓘ discrete Schrödinger operators ⓘ eigenvalue computation ⓘ moment problems ⓘ quantum mechanics models ⓘ three-term recurrence relations ⓘ vibrational analysis of lattices ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Jacobi matrix Description of subject: A Jacobi matrix is a tridiagonal matrix, often symmetric, that arises in numerical analysis and mathematical physics, particularly in the study of orthogonal polynomials and eigenvalue problems.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.