Hurwitz determinants

E714448

determinant mathematical concept stability criterion component tool in control theory

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.

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All labels observed (3)

Label	Occurrences
Hurwitz determinants canonical	1
Hurwitz determinants for higher-degree polynomials	1
Hurwitz polynomial	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.
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
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 ⓘ linked to: Hurwitz matrix
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 ⓘ
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 ⓘ linked to: Hurwitz determinants Lyapunov stability ⓘ linked to: Lyapunov stability theory Routh array ⓘ linked to: Routh–Hurwitz stability criterion 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 ⓘ automatic control ⓘ control theory ⓘ linear systems theory ⓘ signal processing ⓘ

How these facts were elicited

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Routh–Hurwitz stability criterion → uses → Hurwitz determinants ⓘ

Hurwitz determinants → relatedTo → Hurwitz polynomial ⓘ

linked to: Hurwitz determinants

Hurwitz matrix → generalizationOf → Hurwitz determinants for higher-degree polynomials ⓘ