Hurwitz matrix

E714449

concept in control theory concept in linear systems theory matrix structured matrix

The Hurwitz matrix is a structured matrix constructed from the coefficients of a polynomial and used to determine system stability in control theory via the Routh–Hurwitz criterion.

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All labels observed (3)

Label	Occurrences
Hurwitz matrix canonical	1
Hurwitz matrix of the polynomial	1
Hurwitz stability matrix	1

How this entity was disambiguated

This entity first appeared as the object of triple T8119312 — 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 matrix
Context triple: [Routh–Hurwitz stability criterion, relatedConcept, Hurwitz matrix]

A. Sylvester matrix
The Sylvester matrix is a structured matrix constructed from the coefficients of two polynomials, commonly used to compute their resultant and study common roots in algebra.
B. Routh–Hurwitz stability criterion
The Routh–Hurwitz stability criterion is a mathematical test in control theory that determines whether all roots of a system’s characteristic polynomial lie in the left half of the complex plane, ensuring system stability without explicitly computing the roots.
C. Hermite–Biehler theorem
The Hermite–Biehler theorem is a result in complex analysis and control theory that characterizes when a complex polynomial has all its zeros in the open upper half-plane in terms of the interlacing of zeros of two associated real polynomials.
D. 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.
E. 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.
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 matrix
Target entity description: The Hurwitz matrix is a structured matrix constructed from the coefficients of a polynomial and used to determine system stability in control theory via the Routh–Hurwitz criterion.

A. Sylvester matrix
The Sylvester matrix is a structured matrix constructed from the coefficients of two polynomials, commonly used to compute their resultant and study common roots in algebra.
B. Routh–Hurwitz stability criterion
The Routh–Hurwitz stability criterion is a mathematical test in control theory that determines whether all roots of a system’s characteristic polynomial lie in the left half of the complex plane, ensuring system stability without explicitly computing the roots.
C. Hermite–Biehler theorem
The Hermite–Biehler theorem is a result in complex analysis and control theory that characterizes when a complex polynomial has all its zeros in the open upper half-plane in terms of the interlacing of zeros of two associated real polynomials.
D. 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.
E. 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.
F. None of above. chosen

Statements (48)

Predicate	Object
instanceOf	concept in control theory ⓘ concept in linear systems theory ⓘ matrix ⓘ structured matrix ⓘ
appearsIn	classical control theory textbooks ⓘ courses on linear systems and control ⓘ
appliesTo	characteristic polynomials of dynamical systems ⓘ real-coefficient polynomials ⓘ
associatedWith	linear time-invariant differential equations ⓘ transfer functions in control systems ⓘ
basedOn	coefficients of a polynomial ⓘ
constructedFrom	even-indexed coefficients of a polynomial ⓘ odd-indexed coefficients of a polynomial ⓘ ordered coefficients of a polynomial ⓘ
criterionFor	all roots of a polynomial having negative real parts ⓘ
describesProperty	Hurwitz stability of a polynomial ⓘ
field	applied mathematics ⓘ control engineering ⓘ systems and control ⓘ
generalizationOf	Hurwitz determinants for higher-degree polynomials ⓘ linked to: Hurwitz determinants
hasAlternativeName	Hurwitz stability matrix ⓘ linked to: Hurwitz matrix
hasProperty	entries are polynomial coefficients or zeros ⓘ principal minors encode stability information ⓘ size depends on the degree of the polynomial ⓘ
hasPurpose	to determine stability of a linear system ⓘ to test whether all roots of a polynomial lie in the open left half-plane ⓘ
matrixType	Toeplitz-like matrix ⓘ square matrix ⓘ
namedAfter	Adolf Hurwitz ⓘ
relatedConcept	Jury stability criterion ⓘ linked to: Jury test Nyquist stability criterion ⓘ Routh–Hurwitz theorem ⓘ linked to: Routh–Hurwitz stability criterion root locus method ⓘ
relatedTo	Hurwitz determinant ⓘ Hurwitz stability criterion ⓘ linked to: Routh–Hurwitz stability criterion Lyapunov stability ⓘ linked to: Lyapunov stability theory Routh array ⓘ linked to: Routh–Hurwitz stability criterion characteristic equation of a linear system ⓘ
usedBy	applied mathematicians ⓘ control engineers ⓘ systems theorists ⓘ
usedFor	algebraic stability tests without computing roots ⓘ checking necessary and sufficient conditions for stability ⓘ computing Hurwitz determinants ⓘ
usedIn	Routh–Hurwitz stability criterion ⓘ control theory ⓘ polynomial stability tests ⓘ stability analysis of linear time-invariant systems ⓘ

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 → relatedConcept → Hurwitz matrix ⓘ

Hurwitz determinants → constructedFrom → Hurwitz matrix of the polynomial ⓘ

linked to: Hurwitz matrix

Hurwitz matrix → hasAlternativeName → Hurwitz stability matrix ⓘ