Cauchy matrix

In mathematics, a Cauchy matrix, named after Augustin-Louis Cauchy, is an m×n matrix with elements aij in the form



a_{ij}={\frac{1}{x_i-y_j}};\quad x_i-y_j\neq 0,\quad 1 \le i \le m,\quad 1 \le j \le n $$

where $$x_i$$ and $$y_j$$ are elements of a field $$\mathcal{F}$$, and $$(x_i)$$ and $$(y_j)$$ are injective sequences (they contain distinct elements).

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
 * $$ \det \mathbf{A}={{\prod_{i=2}^n \prod_{j=1}^{i-1} (x_i-x_j)(y_j-y_i)}\over {\prod_{i=1}^n \prod_{j=1}^n (x_i-y_j)}}$$ &emsp;&emsp;&emsp;&emsp;(Schechter 1959, eqn 4; Cauchy 1841, p. 154, eqn. 10).

It is always nonzero, and thus all square Cauchy matrices are invertible. The inverse A−1 = B = [bij] is given by
 * $$b_{ij} = (x_j - y_i) A_j(y_i) B_i(x_j) \,$$ &emsp;&emsp;&emsp;&emsp;(Schechter 1959, Theorem 1)

where Ai(x) and Bi(x) are the Lagrange polynomials for $$(x_i)$$ and $$(y_j)$$, respectively. That is,
 * $$A_i(x) = \frac{A(x)}{A^\prime(x_i)(x-x_i)} \quad\text{and}\quad B_i(x) = \frac{B(x)}{B^\prime(y_i)(x-y_i)}, $$

with
 * $$A(x) = \prod_{i=1}^n (x-x_i) \quad\text{and}\quad B(x) = \prod_{i=1}^n (x-y_i). $$

Generalization
A matrix C is called Cauchy-like if it is of the form


 * $$C_{ij}=\frac{r_i s_j}{x_i-y_j}.$$

Defining X=diag(xi), Y=diag(yi), one sees that both Cauchy and Cauchy-like matrices satisfy the displacement equation


 * $$\mathbf{XC}-\mathbf{CY}=rs^\mathrm{T}$$

(with $$r=s=(1,1,\ldots,1)$$ for the Cauchy one). Hence Cauchy-like matrices have a common displacement structure, which can be exploited while working with the matrix. For example, there are known algorithms in literature for Here $$n$$ denotes the size of the matrix (one usually deals with square matrices, though all algorithms can be easily generalized to rectangular matrices).
 * approximate Cauchy matrix-vector multiplication with $$O(n \log n)$$ ops (e.g. the fast multipole method),
 * (pivoted) LU factorization with $$O(n^2)$$ ops (GKO algorithm), and thus linear system solving,
 * approximated or unstable algorithms for linear system solving in $$O(n \log^2 n)$$.