Hermitian forms (work on quadratic forms)
E502192
Hermitian forms (work on quadratic forms) are a class of complex-valued quadratic forms that are linear in one variable and conjugate-linear in the other, generalizing real symmetric quadratic forms and playing a central role in linear algebra and functional analysis.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hermitian forms (work on quadratic forms) canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5191845 — 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: Hermitian forms (work on quadratic forms) Context triple: [Charles Hermite, knownFor, Hermitian forms (work on quadratic forms)]
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A.
Über die Bildung des Formensystems der ternären biquadratischen Form
"Über die Bildung des Formensystems der ternären biquadratischen Form" is the 1907 doctoral dissertation of mathematician Emmy Noether, in which she investigates the invariant theory of certain higher-degree algebraic forms.
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B.
Hilbert’s seventeenth problem
Hilbert’s seventeenth problem is a famous question in real algebraic geometry asking whether every nonnegative polynomial can be represented as a sum of squares of rational functions.
-
C.
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.
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D.
Jacobi’s four-square theorem
Jacobi’s four-square theorem is a fundamental result in number theory that gives a precise formula for the number of ways an integer can be expressed as a sum of four squares.
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E.
Theorie der binären algebraischen Formen
"Theorie der binären algebraischen Formen" is a foundational 19th-century mathematical treatise by Alfred Clebsch on the theory of binary algebraic forms and invariants.
- 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: Hermitian forms (work on quadratic forms) Target entity description: Hermitian forms (work on quadratic forms) are a class of complex-valued quadratic forms that are linear in one variable and conjugate-linear in the other, generalizing real symmetric quadratic forms and playing a central role in linear algebra and functional analysis.
-
A.
Über die Bildung des Formensystems der ternären biquadratischen Form
"Über die Bildung des Formensystems der ternären biquadratischen Form" is the 1907 doctoral dissertation of mathematician Emmy Noether, in which she investigates the invariant theory of certain higher-degree algebraic forms.
-
B.
Hilbert’s seventeenth problem
Hilbert’s seventeenth problem is a famous question in real algebraic geometry asking whether every nonnegative polynomial can be represented as a sum of squares of rational functions.
-
C.
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.
-
D.
Jacobi’s four-square theorem
Jacobi’s four-square theorem is a fundamental result in number theory that gives a precise formula for the number of ways an integer can be expressed as a sum of four squares.
-
E.
Theorie der binären algebraischen Formen
"Theorie der binären algebraischen Formen" is a foundational 19th-century mathematical treatise by Alfred Clebsch on the theory of binary algebraic forms and invariants.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
quadratic form ⓘ sesquilinear form ⓘ |
| additivityCondition | h(x+y,z) = h(x,z) + h(y,z) ⓘ |
| appliesTo |
finite-dimensional complex vector spaces
ⓘ
infinite-dimensional Hilbert spaces ⓘ |
| associatedQuadraticForm | q(x) = h(x,x) ⓘ |
| associatedQuadraticFormProperty | q(x) is real-valued ⓘ |
| classificationInvariants |
nullity
ⓘ
rank ⓘ signature in the indefinite case ⓘ |
| conjugateHomogeneityCondition | h(x,αy) = overline(α) h(x,y) ⓘ |
| conjugateSymmetryCondition | h(x,y) = overline(h(y,x)) ⓘ |
| definedOver | complex vector space ⓘ |
| determines | Hermitian matrix ⓘ |
| diagonalizationProperty | Hermitian matrix can be unitarily diagonalized ⓘ |
| eigenvalueProperty | associated Hermitian matrix has real eigenvalues ⓘ |
| fieldOfStudy |
complex geometry
ⓘ
functional analysis ⓘ linear algebra ⓘ operator theory ⓘ |
| generalizes |
real quadratic form
ⓘ
real symmetric bilinear form ⓘ |
| hasProperty |
conjugate symmetric
ⓘ
conjugate-linear in second argument ⓘ linear in first argument ⓘ |
| hasVariant |
degenerate Hermitian form
ⓘ
indefinite Hermitian form ⓘ positive definite Hermitian form ⓘ |
| historicalOrigin | generalization of symmetric bilinear forms to complex fields ⓘ |
| homogeneityCondition | h(αx,y) = α h(x,y) ⓘ |
| invariantUnder | unitary change of basis ⓘ |
| nameOrigin | named after Charles Hermite ⓘ |
| nondegeneracyCondition | h(x,y)=0 for all y implies x=0 ⓘ |
| orthogonalityCondition | x ⟂ y iff h(x,y)=0 ⓘ |
| relatedConcept |
Hermitian operator
ⓘ
complex inner product ⓘ sesquilinear form ⓘ unitary operator ⓘ |
| representedBy | Hermitian matrix with respect to a basis ⓘ |
| specialCase | inner product when positive definite ⓘ |
| usedFor |
classification of complex quadratic forms
ⓘ
defining inner product on complex vector spaces ⓘ defining norms on complex vector spaces ⓘ defining orthogonality ⓘ geometry of complex projective spaces ⓘ spectral theory of normal operators ⓘ unitary diagonalization of matrices ⓘ |
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: Hermitian forms (work on quadratic forms) Description of subject: Hermitian forms (work on quadratic forms) are a class of complex-valued quadratic forms that are linear in one variable and conjugate-linear in the other, generalizing real symmetric quadratic forms and playing a central role in linear algebra and functional analysis.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.