Cauchy matrix

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

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

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

Properties

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

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

$x_i-y_j=i+j-1.$

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

$det𝐀=\frac{\prod _{i=2}^n\prod _{j=1}^{i-1}(x_i-x_j)(y_j-y_i)}{\prod _{i=1}^n\prod _{j=1}^n(x_i-y_j)}$ (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

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

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

with

$A(x)=\prod _{i=1}^n(x-x_i)\text{and}B(x)=\prod _{i=1}^n(x-y_i).$

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

$𝐗𝐂-𝐂𝐘=r{s}^{\mathrm{T}}$

(with $r=s=(1,1,\dots ,1)$ 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,
• linear system solving in $\tilde{O}({\alpha }^{\omega -1}n)$ ops with the use of fast matrix multiplication algorithms, instead of $\tilde{O}({\alpha }^{2}n)$ ops without it, where $\alpha$ is the displacement rank and ${}^{\sim }2.37\le \omega <3$.
• 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
• Fay's trisecant identity

References

Sources

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