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.

All labels observed (2)

Label Occurrences
Cauchy determinant canonical 1
determinant de Cauchy 1

How this entity was disambiguated

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

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