Cauchy determinant

E239296

determinant formula result in linear algebra

The Cauchy determinant is a classical determinant formula in linear algebra that gives a closed-form expression for matrices with entries of the form 1/(x_i + y_j), named after the French mathematician Augustin-Louis Cauchy.

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All labels observed (2)

Label	Occurrences
Cauchy determinant canonical	1
determinant de Cauchy	1

Statements (31)

Predicate	Object
instanceOf	determinant formula ⓘ result in linear algebra ⓘ
appearsIn	algebraic combinatorics ⓘ classical invariant theory ⓘ
appliesTo	Cauchy matrix ⓘ
assumes	x_i + y_j ≠ 0 for all i,j ⓘ x_i are pairwise distinct ⓘ y_j are pairwise distinct ⓘ
describes	determinant of Cauchy matrix ⓘ
era	19th century mathematics ⓘ
field	linear algebra ⓘ matrix theory ⓘ
generalizationOf	determinant of Vandermonde-type matrices ⓘ
gives	closed-form expression for determinant ⓘ
hasClosedForm	product over i<k (x_k - x_i) times product over j<ℓ (y_ℓ - y_j) divided by product over i,j (x_i + y_j) ⓘ
matrixEntryForm	1/(x_i + y_j) ⓘ 1/(x_i - y_j) ⓘ
namedAfter	Augustin-Louis Cauchy ⓘ
namedInLanguage	Cauchy determinant self-linksurface differs ⓘ surface form: determinant de Cauchy
property	determinant factors into product of differences of x_i and y_j ⓘ determinant is rational function of x_i and y_j ⓘ determinant nonzero if x_i and y_j satisfy distinctness conditions ⓘ
relatedTo	Cauchy matrix ⓘ Cauchy–Binet formula ⓘ Vandermonde matrix ⓘ surface form: Vandermonde determinant
usedFor	computing determinants of structured matrices ⓘ
usedIn	combinatorics ⓘ interpolation theory ⓘ random matrix theory ⓘ representation theory ⓘ theory of special functions ⓘ

How these facts were elicited

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Instruction

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# Requirements
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Input

Subject: Cauchy determinant
Description of subject: The Cauchy determinant is a classical determinant formula in linear algebra that gives a closed-form expression for matrices with entries of the form 1/(x_i + y_j), named after the French mathematician Augustin-Louis Cauchy.

Referenced by (2)

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

Augustin-Louis Cauchy → knownFor → Cauchy determinant ⓘ

Cauchy determinant → namedInLanguage → Cauchy determinant self-linksurface differs ⓘ

this entity surface form: determinant de Cauchy