Cauchy matrix について

are injective sequences (they do not contain repeated elements; elements are ''distinct'').
The Hilbert matrix is a special case of the Cauchy matrix, where
:

x_i-y_j = i+j-1. \;

Every submatrix of a Cauchy matrix is itself a Cauchy matrix.
== Cauchy determinants ==
The determinant of a Cauchy matrix is clearly a rational fraction in the parameters

(x_i)

and

(y_j)

. If the sequences were not injective, the determinant would vanish, and tends to infinity if some

x_i

tends to

y_j

. A subset of its zeros and poles are thus known. The fact is that there are no more zeros and poles:
The determinant of a square Cauchy matrix A is known as a Cauchy determinant and can be given explicitly as
:

\det \mathbf=

(Schechter 1959, eqn 4).
It is always nonzero, and thus all square Cauchy matrices are invertible. The inverse A⁻¹ = B = () is given by
:

b_ = (x_j - y_i) A_j(y_i) B_i(x_j) \,

(Schechter 1959, Theorem 1)
where ''A''_i(x) and ''B''_i(x) are the Lagrange polynomials for

(x_i)

and

(y_j)

, respectively. That is,
:

A_i(x) = \frac \quad\text\quad B_i(x) = \frac,

with
:

A(x) = \prod_^n (x-x_i) \quad\text\quad B(x) = \prod_^n (x-y_i).

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