Hadamard matrices
E334040
Hadamard matrices are square matrices with entries ±1 whose rows are mutually orthogonal, playing a key role in combinatorics, coding theory, and signal processing.
All labels observed (6)
| Label | Occurrences |
|---|---|
| Hadamard conjecture | 1 |
| Hadamard matrices canonical | 1 |
| Hadamard matrix | 1 |
| Hadamard’s determinant problem | 1 |
| Sylvester Hadamard matrix | 1 |
| Walsh matrix | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3167250 — 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: Hadamard matrices Context triple: [Jacques Hadamard, knownFor, Hadamard matrices]
-
A.
Cauchy matrix
A Cauchy matrix is a structured matrix whose entries are defined by the reciprocals of pairwise differences of two sequences, widely used in numerical analysis, interpolation, and algebra.
-
B.
Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
-
C.
Hilbert spaces
Hilbert spaces are complete inner product spaces that provide the fundamental framework for modern functional analysis and many areas of mathematical physics.
-
D.
The Classical Groups: Their Invariants and Representations
The Classical Groups: Their Invariants and Representations is a foundational mathematical monograph by Hermann Weyl that systematically develops the theory of classical Lie groups, their invariants, and their representation theory.
-
E.
Kailath factorization in linear systems
Kailath factorization in linear systems is a matrix factorization technique used in control and signal processing to efficiently analyze and solve linear dynamical systems.
- 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: Hadamard matrices Target entity description: Hadamard matrices are square matrices with entries ±1 whose rows are mutually orthogonal, playing a key role in combinatorics, coding theory, and signal processing.
-
A.
Cauchy matrix
A Cauchy matrix is a structured matrix whose entries are defined by the reciprocals of pairwise differences of two sequences, widely used in numerical analysis, interpolation, and algebra.
-
B.
Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
-
C.
Hilbert spaces
Hilbert spaces are complete inner product spaces that provide the fundamental framework for modern functional analysis and many areas of mathematical physics.
-
D.
The Classical Groups: Their Invariants and Representations
The Classical Groups: Their Invariants and Representations is a foundational mathematical monograph by Hermann Weyl that systematically develops the theory of classical Lie groups, their invariants, and their representation theory.
-
E.
Kailath factorization in linear systems
Kailath factorization in linear systems is a matrix factorization technique used in control and signal processing to efficiently analyze and solve linear dynamical systems.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
combinatorial object
ⓘ
matrix class ⓘ orthogonal matrix over ±1 ⓘ |
| constructedBy | Kronecker product recursion ⓘ |
| fieldOfStudy |
coding theory
ⓘ
combinatorics ⓘ design theory ⓘ linear algebra ⓘ |
| hasConjecture |
Hadamard matrices
self-linksurface differs
ⓘ
surface form:
Hadamard conjecture
|
| hasConstraintOnOrder | n = 1, 2, or a multiple of 4 ⓘ |
| hasConstructionMethod |
Goethals–Seidel construction
ⓘ
Paley construction ⓘ Sylvester construction ⓘ Williamson construction ⓘ tensor (Kronecker) product of smaller Hadamard matrices ⓘ |
| hasEntrySet | {+1, -1} ⓘ |
| hasEquivalentFormulation | matrix with entries ±1 whose rows have pairwise Hamming distance n/2 after mapping ±1 to {0,1} ⓘ |
| hasOrder |
2^k for integer k ≥ 0
ⓘ
n where n is a positive integer ⓘ |
| hasProperty |
columns form an orthogonal set over the reals
ⓘ
columns mutually orthogonal ⓘ determinant has maximal absolute value among ±1 matrices of same order ⓘ entries of each row have equal magnitude ⓘ rows form an orthogonal set over the reals ⓘ rows mutually orthogonal ⓘ square matrix ⓘ |
| hasSpecialCase |
Hadamard matrices
self-linksurface differs
ⓘ
surface form:
Sylvester Hadamard matrix
|
| maximizes | determinant among ±1 matrices of given order ⓘ |
| namedAfter | Jacques Hadamard ⓘ |
| relatedTo |
Hadamard inequality
ⓘ
Hadamard matrices self-linksurface differs ⓘ
surface form:
Walsh matrix
binary linear code ⓘ conference matrix ⓘ orthogonal array ⓘ |
| satisfies | H H^T = n I_n for an n×n Hadamard matrix H ⓘ |
| states | Hadamard matrices exist for every order n that is a multiple of 4 ⓘ |
| usedIn |
Walsh–Hadamard transform
ⓘ
coding theory ⓘ combinatorial design theory ⓘ construction of Hadamard codes ⓘ construction of Reed–Muller codes ⓘ construction of mutually unbiased bases ⓘ construction of orthogonal arrays ⓘ construction of symmetric balanced incomplete block designs ⓘ error-correcting codes ⓘ experimental design ⓘ orthogonal frequency-division multiplexing variants ⓘ quantum information theory ⓘ signal processing ⓘ spectral methods in combinatorics ⓘ |
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: Hadamard matrices Description of subject: Hadamard matrices are square matrices with entries ±1 whose rows are mutually orthogonal, playing a key role in combinatorics, coding theory, and signal processing.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Hadamard matrix
this entity surface form:
Hadamard conjecture
subject surface form:
Hadamard matrix
this entity surface form:
Sylvester Hadamard matrix
subject surface form:
Hadamard matrix
this entity surface form:
Walsh matrix
this entity surface form:
Hadamard matrix
this entity surface form:
Hadamard’s determinant problem