Hurwitz determinants
E714448
Hurwitz determinants are specific determinants constructed from a polynomial’s coefficients that are used to test whether all roots of the polynomial lie in the left half of the complex plane, thereby assessing system stability.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hurwitz determinants canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T8119295 — 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: Hurwitz determinants Context triple: [Routh–Hurwitz stability criterion, uses, Hurwitz determinants]
-
A.
Hurwitz theorem
Hurwitz theorem is a fundamental result in Diophantine approximation that gives an optimal bound on how well any irrational real number can be approximated by infinitely many rational numbers.
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B.
Sylvester determinant
The Sylvester determinant is a mathematical construct introduced by James Joseph Sylvester, typically referring to a determinant associated with resultants and elimination theory in algebra.
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C.
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.
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D.
Hurwitz bound on automorphism groups of curves
The Hurwitz bound on automorphism groups of curves is a classical result in algebraic geometry stating that a compact Riemann surface of genus at least 2 has at most 84(g − 1) automorphisms.
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E.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
- 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: Hurwitz determinants Target entity description: Hurwitz determinants are specific determinants constructed from a polynomial’s coefficients that are used to test whether all roots of the polynomial lie in the left half of the complex plane, thereby assessing system stability.
-
A.
Hurwitz theorem
Hurwitz theorem is a fundamental result in Diophantine approximation that gives an optimal bound on how well any irrational real number can be approximated by infinitely many rational numbers.
-
B.
Sylvester determinant
The Sylvester determinant is a mathematical construct introduced by James Joseph Sylvester, typically referring to a determinant associated with resultants and elimination theory in algebra.
-
C.
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.
-
D.
Hurwitz bound on automorphism groups of curves
The Hurwitz bound on automorphism groups of curves is a classical result in algebraic geometry stating that a compact Riemann surface of genus at least 2 has at most 84(g − 1) automorphisms.
-
E.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
determinant
ⓘ
mathematical concept ⓘ stability criterion component ⓘ tool in control theory ⓘ |
| advantage | avoid explicit computation of polynomial roots ⓘ |
| appliesTo |
characteristic polynomials of linear time-invariant systems
ⓘ
continuous-time linear systems ⓘ univariate polynomials with real coefficients ⓘ |
| assumes |
polynomial degree is finite
ⓘ
polynomial has real coefficients ⓘ |
| basedOn | coefficients of a polynomial ⓘ |
| condition | all leading principal minors of the Hurwitz matrix must be positive for stability ⓘ |
| constructedFrom | Hurwitz matrix of the polynomial NERFINISHED ⓘ |
| criterionFor |
all roots having negative real parts
ⓘ
location of polynomial zeros in the complex plane ⓘ |
| field |
algebra
ⓘ
applied mathematics ⓘ complex analysis ⓘ systems and control ⓘ |
| generalizationOf | stability tests based on principal minors ⓘ |
| influenced | development of algebraic stability criteria in control theory ⓘ |
| introducedIn | 19th century ⓘ |
| mathematicalNature | leading principal minors of a structured matrix ⓘ |
| namedAfter | Adolf Hurwitz NERFINISHED ⓘ |
| notTypicallyUsedFor | discrete-time stability in the z-plane ⓘ |
| propertyTested |
Hurwitz stability
ⓘ
location of roots relative to the imaginary axis ⓘ |
| purpose |
assess asymptotic stability of linear time-invariant systems
ⓘ
check Hurwitz stability of a polynomial ⓘ test whether all roots of a polynomial lie in the open left half-plane ⓘ |
| relatedTo |
Hurwitz matrix
ⓘ
Hurwitz polynomial NERFINISHED ⓘ Lyapunov stability NERFINISHED ⓘ Routh array NERFINISHED ⓘ characteristic equation ⓘ eigenvalues of system matrix ⓘ |
| requires | arranging polynomial coefficients in a specific structured matrix ⓘ |
| usedFor |
design and analysis of feedback control systems
ⓘ
stability margins assessment ⓘ verification of closed-loop stability without computing roots explicitly ⓘ |
| usedIn |
Hurwitz stability criterion
ⓘ
Routh–Hurwitz stability criterion NERFINISHED ⓘ automatic control ⓘ control theory ⓘ linear systems theory ⓘ signal processing ⓘ |
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: Hurwitz determinants Description of subject: Hurwitz determinants are specific determinants constructed from a polynomial’s coefficients that are used to test whether all roots of the polynomial lie in the left half of the complex plane, thereby assessing system stability.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.