Sylvester matrix

E571002

mathematical object matrix structured 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.

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Statements (45)

Predicate	Object
instanceOf	mathematical object ⓘ matrix ⓘ structured matrix ⓘ
appearsIn	eigenvalue methods for polynomial equations ⓘ resultant-based polynomial system solvers ⓘ
appliesTo	multivariate polynomials via specialization ⓘ univariate polynomials ⓘ
constructionMethod	rows formed from shifted coefficient vectors of the first polynomial ⓘ rows formed from shifted coefficient vectors of the second polynomial ⓘ
countryOfOrigin	United Kingdom ⓘ
dimensionRule	for degrees m and n the Sylvester matrix is of size (m+n)×(m+n) ⓘ
field	algebra ⓘ linear algebra ⓘ
generalization	block Sylvester matrix ⓘ multivariate Sylvester matrix ⓘ
hasProperty	block Toeplitz structure ⓘ can be ill-conditioned for high-degree polynomials ⓘ depends linearly on polynomial coefficients ⓘ determinant equals the resultant of the polynomials ⓘ rectangular in general ⓘ square when constructed from two univariate polynomials of fixed degrees ⓘ
implies	vanishing determinant if and only if polynomials have a common root ⓘ
namedAfter	James Joseph Sylvester NERFINISHED ⓘ
relatedTo	Bezout matrix NERFINISHED ⓘ Sylvester equation NERFINISHED ⓘ Sylvester resultant NERFINISHED ⓘ companion matrix ⓘ discriminant of a polynomial ⓘ resultant ⓘ
symbol	often denoted S(f,g) for polynomials f and g ⓘ
typicalInput	coefficients of two polynomials ⓘ
usedFor	computing polynomial greatest common divisors ⓘ computing the resultant of two polynomials ⓘ elimination theory ⓘ resultant-based root finding ⓘ studying common roots of two polynomials ⓘ testing whether two polynomials have a common root ⓘ
usedIn	algebraic geometry ⓘ computational algebra ⓘ computer algebra systems ⓘ control theory ⓘ numerical linear algebra ⓘ signal processing ⓘ symbolic computation ⓘ systems theory ⓘ

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James Joseph Sylvester → notableWork → Sylvester matrix ⓘ