Cauchy matrix

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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 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[edit]

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)}}     (Schechter 1959, eqn 4).

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) \,     (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[edit]

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

  • 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).

Here n denotes the size of the matrix (one usually deals with square matrices, though all algorithms can be easily generalized to rectangular matrices).

See also[edit]

References[edit]