Vandermonde matrix

E620657

matrix structured matrix

A Vandermonde matrix is a structured matrix whose rows (or columns) are geometric progressions of given numbers, widely used in polynomial interpolation, determinant theory, and numerical analysis.

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Observed surface forms (1)

Surface form	Occurrences
Vandermonde determinant	2

Statements (47)

Predicate	Object
instanceOf	matrix ⓘ structured matrix ⓘ
appearsIn	Prony’s method NERFINISHED ⓘ barycentric interpolation formulas ⓘ exponential fitting ⓘ moment problems ⓘ
belongsTo	class of polynomial evaluation matrices ⓘ
field	determinant theory ⓘ linear algebra ⓘ numerical analysis ⓘ polynomial interpolation ⓘ
hasAlgorithm	divide-and-conquer methods for Vandermonde systems ⓘ specialized O(n^2) algorithms for solving Vandermonde systems ⓘ
hasAlternativeBasis	can be replaced by orthogonal polynomial bases for better conditioning ⓘ
hasColumnForm	(1, x_j, x_j^2, …, x_j^{m-1})^T for given scalars x_j ⓘ
hasComputationIssue	direct solution is numerically unstable for large systems ⓘ
hasDeterminantFormula	det(V) = ∏_{1 ≤ i < j ≤ n} (x_j − x_i) ⓘ
hasGeneralization	block Vandermonde matrix NERFINISHED ⓘ confluent Vandermonde matrix NERFINISHED ⓘ q-Vandermonde matrix NERFINISHED ⓘ
hasProperty	can be expressed as evaluation matrix of monomials at nodes ⓘ columns form powers of the nodes x_i ⓘ determinant is zero iff some x_i = x_j for i ≠ j ⓘ ill-conditioned for large n or clustered nodes ⓘ rows form geometric progressions ⓘ
hasRowForm	(1, x_i, x_i^2, …, x_i^{n-1}) for given scalars x_i ⓘ
hasSize	m×n for m rows and n columns ⓘ
hasSpecialCase	DFT matrix when nodes are roots of unity ⓘ
isDefinedOver	any field ⓘ complex numbers ⓘ real numbers ⓘ
isNonsingularIf	all x_i are pairwise distinct ⓘ
isSingularIf	two or more x_i coincide ⓘ
namedAfter	Alexandre-Théophile Vandermonde NERFINISHED ⓘ
relatedTo	Cauchy matrix NERFINISHED ⓘ Lagrange interpolation NERFINISHED ⓘ Newton interpolation NERFINISHED ⓘ Toeplitz matrix NERFINISHED ⓘ companion matrix ⓘ
usedFor	coding theory ⓘ computing coefficients of interpolating polynomials ⓘ determinant evaluation ⓘ discrete Fourier transform generalizations ⓘ polynomial interpolation ⓘ signal processing ⓘ solving systems with polynomial data ⓘ system identification ⓘ

Referenced by (4)

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

Lagrange interpolation polynomial → relatedTo → Vandermonde matrix ⓘ

Cauchy matrix → relatedTo → Vandermonde matrix ⓘ

Cauchy determinant → relatedTo → Vandermonde matrix ⓘ

this entity surface form: Vandermonde determinant

Selberg integral → relatedTo → Vandermonde matrix ⓘ

this entity surface form: Vandermonde determinant