Cauchy determinant

E239296

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

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Instruction
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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