Kronecker product
E102485
The Kronecker product is a matrix operation that forms a large block matrix from two smaller matrices and is widely used in linear algebra, quantum computing, and signal processing.
All labels observed (2)
| Label | Occurrences |
|---|---|
| For an m×n matrix A and a p×q matrix B, A ⊗ B is the mp×nq block matrix whose (i,j)-th block is a_ij B | 1 |
| Kronecker product canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T846898 — 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: Kronecker product Context triple: [Leopold Kronecker, notableWork, Kronecker product]
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A.
Kronecker delta
The Kronecker delta is a function of two variables that equals 1 when the variables are equal and 0 otherwise, widely used in linear algebra, tensor calculus, and discrete mathematics to represent identity relations.
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B.
Kronecker’s lemma
Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
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C.
LinearAlgebra
LinearAlgebra is Julia’s standard library module providing core functionality for vectors, matrices, and advanced linear algebra operations.
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D.
Matrix
Matrix is a professional haircare and hair color brand widely used in salons and owned by the cosmetics company L'Oréal.
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E.
Matrix
The Matrix is a vast, sentient data repository and virtual reality construct on Gallifrey that stores the knowledge, memories, and consciousness of Time Lords in the Doctor Who universe.
- 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: Kronecker product Target entity description: The Kronecker product is a matrix operation that forms a large block matrix from two smaller matrices and is widely used in linear algebra, quantum computing, and signal processing.
-
A.
Kronecker delta
The Kronecker delta is a function of two variables that equals 1 when the variables are equal and 0 otherwise, widely used in linear algebra, tensor calculus, and discrete mathematics to represent identity relations.
-
B.
Kronecker’s lemma
Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
-
C.
LinearAlgebra
LinearAlgebra is Julia’s standard library module providing core functionality for vectors, matrices, and advanced linear algebra operations.
-
D.
Matrix
Matrix is a professional haircare and hair color brand widely used in salons and owned by the cosmetics company L'Oréal.
-
E.
Matrix
The Matrix is a vast, sentient data repository and virtual reality construct on Gallifrey that stores the knowledge, memories, and consciousness of Time Lords in the Doctor Who universe.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
binary operation on matrices
ⓘ
matrix operation ⓘ |
| alsoKnownAs |
direct product of matrices
ⓘ
tensor product of matrices ⓘ |
| arity | 2 ⓘ |
| definition |
For an m×n matrix A and a p×q matrix B, A ⊗ B is the mp×nq block matrix whose (i,j)-th block is a_ij B
ⓘ
linked to:
Kronecker product
|
| domain | finite-dimensional vector spaces ⓘ |
| eigenvalueRelation | Eigenvalues of A ⊗ B are pairwise products of eigenvalues of A and B ⓘ |
| field |
control theory
ⓘ
linear algebra ⓘ matrix theory ⓘ numerical analysis ⓘ quantum computing ⓘ signal processing ⓘ statistics ⓘ |
| identityRelation | I_m ⊗ I_n = I_{mn} ⓘ |
| inputType |
matrix
ⓘ
matrix over a ring ⓘ |
| inverseRelation | If A and B are invertible then (A ⊗ B)^{-1} = A^{-1} ⊗ B^{-1} ⓘ |
| linearity | bilinear in both arguments ⓘ |
| matrixSizeRule | If A is m×n and B is p×q then A ⊗ B is mp×nq ⓘ |
| namedAfter | Leopold Kronecker ⓘ |
| outputType | matrix ⓘ |
| property |
(A ⊗ B)(C ⊗ D) = AC ⊗ BD when dimensions are compatible
ⓘ
(A ⊗ B)^* = A^* ⊗ B^* for conjugate transpose ⓘ (A ⊗ B)^T = A^T ⊗ B^T ⓘ associative up to canonical isomorphism ⓘ compatible with scalar multiplication ⓘ det(A ⊗ B) = det(A)^p det(B)^m for A m×m and B p×p ⓘ distributive over matrix addition ⓘ non-commutative in general ⓘ rank(A ⊗ B) = rank(A) rank(B) ⓘ trace(A ⊗ B) = trace(A) trace(B) ⓘ |
| relatedConcept |
Hadamard product
ⓘ
matrix direct sum ⓘ tensor product of vector spaces ⓘ vec operator ⓘ |
| standardIdentity | vec(AXB) = (B^T ⊗ A) vec(X) when dimensions are compatible ⓘ |
| symbol |
\otimes
ⓘ
⊗ ⓘ |
| use |
construction of large structured matrices
ⓘ
discretization of partial differential equations ⓘ formulation of multi-qubit quantum gates ⓘ modeling composite quantum systems ⓘ multidimensional signal processing ⓘ representation of linear maps on tensor product spaces ⓘ separable covariance matrices in statistics ⓘ solution of large-scale linear systems ⓘ vectorization identities in matrix calculus ⓘ |
| zeroRelation | 0 ⊗ A = 0 and A ⊗ 0 = 0 ⓘ |
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: Kronecker product Description of subject: The Kronecker product is a matrix operation that forms a large block matrix from two smaller matrices and is widely used in linear algebra, quantum computing, and signal processing.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
Kronecker product
→
definition
→
For an m×n matrix A and a p×q matrix B, A ⊗ B is the mp×nq block matrix whose (i,j)-th block is a_ij B
ⓘ
linked to: Kronecker product