Cauchy determinant

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

:$a_=;\quad x_i-y_j\neq 0,\quad 1 \le i \le m,\quad 1 \le j \le n$

where $x_i$ and $y_j$ are elements of a field $\mathcal$, 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
:$\det \mathbf=$     (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 = _ijis given by
:$b_ = (x_j - y_i) A_j(y_i) B_i(x_j) \,$     (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,
:$A_i(x) = \frac \quad\text\quad B_i(x) = \frac,$
with
:$A(x) = \prod_^n (x-x_i) \quad\text\quad B(x) = \prod_^n (x-y_i).$

Generalization

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

:$C_=\frac.$

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

:$\mathbf-\mathbf=rs^\mathrm$

(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
* Fay's trisecant identity

References

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* TiIo Finck, Georg Heinig, and Karla Rost: "An Inversion Formula and Fast Algorithms for Cauchy-Vandermonde Matrices", Linear Algebra and its Applications, vol.183 (1993), pp.179-191.
* Dario Fasino: "Orthogonal Cauchy-like matrices", Numerical Algorithms, vol.92 (2023), pp.619-637. url=https://doi.org/10.1007/s11075-022-01391-y .

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