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).

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

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

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 ${\displaystyle (x_{i})}$ 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

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

Generalization

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,
• 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).

See also

• Toeplitz matrix
• Fay's trisecant identity

References

• Cauchy, Augustin-Louis (1841). Exercices d'analyse et de physique mathématique. Vol. 2 (in French). Bachelier.
• 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. JSTOR 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. Bibcode:1995MaCom..64.1557G. doi:10.1090/s0025-5718-1995-1312096-x.
• P. G. Martinsson; M. Tygert; V. Rokhlin (2005). "An ${\displaystyle O(N\log ^{2}N)}$ algorithm for the inversion of general Toeplitz matrices" (PDF). Computers & Mathematics with Applications. 50 (5–6): 741–752. doi:10.1016/j.camwa.2005.03.011.
• 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. JSTOR 2001955.
• 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 .