Cauchy matrix

In mathematics, a Cauchy matrix, named after Augustin Louis Cauchy, is an m×n matrix with elements a_ij 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

$\det \mathbf{A}={{\prod_{i=2}^n \prod_{j=1}^{i-1} (x_i-x_j)(y_j-y_i)}\over {\prod_{i=1}^n \prod_{j=1}^n (x_i-y_j)}}$     (Schechter 1959, eqn 4).

It is always nonzero, and thus all square Cauchy matrices are invertible. The inverse A⁻¹ = B = [b_ij] is given by

$b_{ij} = (x_j - y_i) A_j(y_i) B_i(x_j) \,$     (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,

$A_i(x) = \frac{A(x)}{A^\prime(x_i)(x-x_i)} \quad\text{and}\quad B_i(x) = \frac{B(x)}{B^\prime(y_i)(x-y_i)},$

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(x_i), Y=diag(y_i), 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

• Toeplitz matrix

References

• A. Gerasoulis (1988). "A fast algorithm for the multiplication of generalized Hilbert matrices with vectors" (PDF). Mathematics of Computation 50 (181): 179–188. doi:10.2307/2007921.
• I. Gohberg, T. Kailath, V. Olshevsky (1995). "Fast Gaussian elimination with partial pivoting for matrices with displacement structure" (PDF). Mathematics of Computation 64 (212): 1557–1576. doi:10.1090/s0025-5718-1995-1312096-x.
• P. G. Martinsson, M. Tygert, V. Rokhlin (2005). $O(N \log^2 N)$ (PDF). Computers & Mathematics with Applications 50: 741–752. doi:10.1016/j.camwa.2005.03.011. delete character in |title= at position 4 (help)
• S. Schechter (1959). "On the inversion of certain matrices" (PDF). Mathematical Tables and Other Aids to Computation 13 (66): 73–77. doi:10.2307/2001955.