Cauchy matrix

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

${\displaystyle a_{ij}={\frac {1}{x_{i}-y_{j}}};\quad x_{i}-y_{j}\neq 0,\quad 1\leq i\leq m,\quad 1\leq j\leq n}$

where ${\displaystyle x_{i}}$ and ${\displaystyle y_{j}}$ are elements of a field ${\displaystyle {\mathcal {F}}}$, and ${\displaystyle (x_{i})}$ and ${\displaystyle (y_{j})}$ are injective sequences (they contain distinct elements).

Every submatrix of a Cauchy matrix is itself a Cauchy matrix.

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

${\displaystyle x_{i}-y_{j}=i+j-1.\;}$

The determinant of a Cauchy matrix is clearly a rational fraction in the parameters ${\displaystyle (x_{i})}$ and ${\displaystyle (y_{j})}$. If the sequences were not injective, the determinant would vanish, and tends to infinity if some ${\displaystyle x_{i}}$ tends to ${\displaystyle y_{j}}$. 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

${\displaystyle \det \mathbf {A} ={{\prod _{i=2}^{n}\prod _{j=1}^{i-1}(x_{i}-x_{j})(y_{j}-y_{i})} \over {\prod _{i=1}^{n}\prod _{j=1}^{n}(x_{i}-y_{j})}}}$ (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 = [b_ij] is given by

${\displaystyle b_{ij}=(x_{j}-y_{i})A_{j}(y_{i})B_{i}(x_{j})\,}$ (Schechter 1959, Theorem 1)

where A_i(x) and B_i(x) are the Lagrange polynomials for ${\displaystyle (x_{i})}$ and ${\displaystyle (y_{j})}$, respectively. That is,

${\displaystyle A_{i}(x)={\frac {A(x)}{A^{\prime }(x_{i})(x-x_{i})}}\quad {\text{and}}\quad B_{i}(x)={\frac {B(x)}{B^{\prime }(y_{i})(x-y_{i})}},}$

with

${\displaystyle A(x)=\prod _{i=1}^{n}(x-x_{i})\quad {\text{and}}\quad B(x)=\prod _{i=1}^{n}(x-y_{i}).}$

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

${\displaystyle C_{ij}={\frac {r_{i}s_{j}}{x_{i}-y_{j}}}.}$

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

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

(with ${\displaystyle r=s=(1,1,\ldots ,1)}$ 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 ${\displaystyle O(n\log n)}$ ops (e.g. the fast multipole method),
• (pivoted) LU factorization with ${\displaystyle O(n^{2})}$ ops (GKO algorithm), and thus linear system solving,
• linear system solving in ${\displaystyle {\tilde {O}}({\alpha ^{\omega -1}}n)}$ ops with the use of fast matrix multiplication algorithms, instead of ${\displaystyle {\tilde {O}}({\alpha ^{2}}n)}$ ops without it, where ${\displaystyle \alpha }$ is the displacement rank and ${\displaystyle ^{\sim }2.37\leq \omega <3}$¹.
• approximated or unstable algorithms for linear system solving in ${\displaystyle O(n\log ^{2}n)}$.

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