Hadamard inequality
E334041
The Hadamard inequality is a fundamental result in linear algebra and analysis that bounds the absolute value of a determinant by the product of the Euclidean norms of its row or column vectors.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Hadamard inequality canonical | 2 |
| Hadamard’s inequality for convex functions | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3167251 — 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 inequality Context triple: [Jacques Hadamard, knownFor, Hadamard inequality]
-
A.
Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality is a fundamental result in linear algebra and analysis that bounds the inner product of two vectors by the product of their magnitudes, underpinning many concepts in geometry, probability, and functional analysis.
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B.
Minkowski inequality
The Minkowski inequality is a fundamental result in functional analysis and measure theory that generalizes the triangle inequality to L^p spaces, providing a key tool for studying norms and integrable functions.
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C.
Hölder inequality
Hölder inequality is a fundamental result in mathematical analysis that generalizes the Cauchy–Schwarz inequality and provides bounds for integrals or sums of products in Lᵖ spaces.
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D.
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.
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E.
Cauchy interlacing theorem
The Cauchy interlacing theorem is a fundamental result in linear algebra that relates the eigenvalues of a symmetric matrix to those of its principal submatrices, showing how they "interlace" on the real line.
- 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 inequality Target entity description: The Hadamard inequality is a fundamental result in linear algebra and analysis that bounds the absolute value of a determinant by the product of the Euclidean norms of its row or column vectors.
-
A.
Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality is a fundamental result in linear algebra and analysis that bounds the inner product of two vectors by the product of their magnitudes, underpinning many concepts in geometry, probability, and functional analysis.
-
B.
Minkowski inequality
The Minkowski inequality is a fundamental result in functional analysis and measure theory that generalizes the triangle inequality to L^p spaces, providing a key tool for studying norms and integrable functions.
-
C.
Hölder inequality
Hölder inequality is a fundamental result in mathematical analysis that generalizes the Cauchy–Schwarz inequality and provides bounds for integrals or sums of products in Lᵖ spaces.
-
D.
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.
-
E.
Cauchy interlacing theorem
The Cauchy interlacing theorem is a fundamental result in linear algebra that relates the eigenvalues of a symmetric matrix to those of its principal submatrices, showing how they "interlace" on the real line.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical inequality
ⓘ
result in analysis ⓘ result in linear algebra ⓘ |
| appliesTo |
complex matrices
ⓘ
real matrices ⓘ square matrices ⓘ |
| category |
determinant inequalities
ⓘ
inequalities in matrix analysis ⓘ |
| centuryOfIntroduction | 19th century ⓘ |
| describes | upper bound for determinant of a matrix ⓘ |
| equalityCondition |
columns are pairwise orthogonal
ⓘ
matrix is diagonal up to unitary transformation ⓘ rows are pairwise orthogonal ⓘ |
| field |
convex analysis
ⓘ
functional analysis ⓘ linear algebra ⓘ matrix theory ⓘ |
| generalizationOf |
bound on area of parallelogram by product of side lengths
ⓘ
bound on volume of parallelotope by product of edge lengths ⓘ |
| hasVariant |
Hadamard three-lines theorem (terminological relation only)
ⓘ
Hadamard inequality self-linksurface differs ⓘ
surface form:
Hadamard’s inequality for convex functions
|
| holdsFor | Hermitian positive semidefinite matrices via Gram representation ⓘ |
| implies |
determinant is maximized by orthogonal columns for fixed column norms
ⓘ
determinant is maximized by orthogonal rows for fixed row norms ⓘ |
| introducedBy | Jacques Hadamard ⓘ |
| namedAfter | Jacques Hadamard ⓘ |
| proofTechnique |
Cauchy–Binet formula
ⓘ
Cauchy–Schwarz inequality ⓘ induction on matrix size ⓘ |
| relatedTo |
Cauchy–Schwarz inequality
ⓘ
Gram determinant ⓘ Hadamard matrices ⓘ
surface form:
Hadamard matrix
Hadamard matrices ⓘ
surface form:
Hadamard’s determinant problem
|
| statementForm |
|det(A)| ≤ ∏‖col_j(A)‖₂
ⓘ
|det(A)| ≤ ∏‖row_i(A)‖₂ ⓘ |
| usedAs | tool to bound determinants in analysis ⓘ |
| usedIn |
estimates for condition numbers
ⓘ
matrix analysis ⓘ numerical linear algebra ⓘ optimization of determinants ⓘ probability theory on random matrices ⓘ |
| usesConcept |
column vectors
ⓘ
determinant ⓘ row vectors ⓘ |
| usesNorm |
Euclidean metric
ⓘ
surface form:
Euclidean norm
|
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 inequality Description of subject: The Hadamard inequality is a fundamental result in linear algebra and analysis that bounds the absolute value of a determinant by the product of the Euclidean norms of its row or column vectors.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Hadamard matrix
this entity surface form:
Hadamard’s inequality for convex functions