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

[math]\displaystyle{ 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 }[/math]

where [math]\displaystyle{ x_i }[/math] and [math]\displaystyle{ y_j }[/math] are elements of a field [math]\displaystyle{ \mathcal{F} }[/math], and [math]\displaystyle{ (x_i) }[/math] and [math]\displaystyle{ (y_j) }[/math] are injective sequences (they contain distinct elements).

The Hilbert matrix is a special case of the Cauchy matrix, where

[math]\displaystyle{ x_i-y_j = i+j-1. \; }[/math]

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 [math]\displaystyle{ (x_i) }[/math] and [math]\displaystyle{ (y_j) }[/math]. If the sequences were not injective, the determinant would vanish, and tends to infinity if some [math]\displaystyle{ x_i }[/math] tends to [math]\displaystyle{ y_j }[/math]. 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

[math]\displaystyle{ \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)}} }[/math]     (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

[math]\displaystyle{ b_{ij} = (x_j - y_i) A_j(y_i) B_i(x_j) \, }[/math]     (Schechter 1959, Theorem 1)

where Ai(x) and Bi(x) are the Lagrange polynomials for [math]\displaystyle{ (x_i) }[/math] and [math]\displaystyle{ (y_j) }[/math], respectively. That is,

[math]\displaystyle{ 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)}, }[/math]

with

[math]\displaystyle{ A(x) = \prod_{i=1}^n (x-x_i) \quad\text{and}\quad B(x) = \prod_{i=1}^n (x-y_i). }[/math]

Generalization

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

[math]\displaystyle{ C_{ij}=\frac{r_i s_j}{x_i-y_j}. }[/math]

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

[math]\displaystyle{ \mathbf{XC}-\mathbf{CY}=rs^\mathrm{T} }[/math]

(with [math]\displaystyle{ r=s=(1,1,\ldots,1) }[/math] 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 [math]\displaystyle{ O(n \log n) }[/math] ops (e.g. the fast multipole method),
  • (pivoted) LU factorization with [math]\displaystyle{ O(n^2) }[/math] ops (GKO algorithm), and thus linear system solving,
  • approximated or unstable algorithms for linear system solving in [math]\displaystyle{ O(n \log^2 n) }[/math].

Here [math]\displaystyle{ n }[/math] denotes the size of the matrix (one usually deals with square matrices, though all algorithms can be easily generalized to rectangular matrices).

See also

References



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