Cauchy matrix

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

[math]\displaystyle{ 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 }[/math]

where [math]\displaystyle{ x_i }[/math] and [math]\displaystyle{ y_j }[/math] are elements of a field [math]\displaystyle{ \mathcal{F} }[/math], and [math]\displaystyle{ (x_i) }[/math] and [math]\displaystyle{ (y_j) }[/math] are injective sequences (they contain distinct elements).

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

[math]\displaystyle{ x_i-y_j = i+j-1. \; }[/math]

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 [math]\displaystyle{ (x_i) }[/math] and [math]\displaystyle{ (y_j) }[/math]. If the sequences were not injective, the determinant would vanish, and tends to infinity if some [math]\displaystyle{ x_i }[/math] tends to [math]\displaystyle{ y_j }[/math]. 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

[math]\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)}} }[/math]     (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

[math]\displaystyle{ b_{ij} = (x_j - y_i) A_j(y_i) B_i(x_j) \, }[/math]     (Schechter 1959, Theorem 1)

where A_i(x) and B_i(x) are the Lagrange polynomials for [math]\displaystyle{ (x_i) }[/math] and [math]\displaystyle{ (y_j) }[/math], respectively. That is,

[math]\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)}, }[/math]

with

[math]\displaystyle{ A(x) = \prod_{i=1}^n (x-x_i) \quad\text{and}\quad B(x) = \prod_{i=1}^n (x-y_i). }[/math]

Generalization

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

[math]\displaystyle{ C_{ij}=\frac{r_i s_j}{x_i-y_j}. }[/math]

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

[math]\displaystyle{ \mathbf{XC}-\mathbf{CY}=rs^\mathrm{T} }[/math]

(with [math]\displaystyle{ r=s=(1,1,\ldots,1) }[/math] 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 [math]\displaystyle{ O(n \log n) }[/math] ops (e.g. the fast multipole method),
• (pivoted) LU factorization with [math]\displaystyle{ O(n^2) }[/math] ops (GKO algorithm), and thus linear system solving,
• approximated or unstable algorithms for linear system solving in [math]\displaystyle{ O(n \log^2 n) }[/math].

Here [math]\displaystyle{ n }[/math] 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) (in fr). Exercices d'analyse et de physique mathématique. Vol. 2. Bachelier. https://books.google.com/books?id=DRg-AQAAIAAJ. 
• A. Gerasoulis (1988). "A fast algorithm for the multiplication of generalized Hilbert matrices with vectors". Mathematics of Computation 50 (181): 179–188. doi:10.2307/2007921. https://www.ams.org/journals/mcom/1988-50-181/S0025-5718-1988-0917825-9/S0025-5718-1988-0917825-9.pdf. 
• I. Gohberg; T. Kailath; V. Olshevsky (1995). "Fast Gaussian elimination with partial pivoting for matrices with displacement structure". Mathematics of Computation 64 (212): 1557–1576. doi:10.1090/s0025-5718-1995-1312096-x. Bibcode: 1995MaCom..64.1557G. https://www.ams.org/journals/mcom/1995-64-212/S0025-5718-1995-1312096-X/S0025-5718-1995-1312096-X.pdf. 
• P. G. Martinsson; M. Tygert; V. Rokhlin (2005). "An [math]\displaystyle{ O(N \log^2 N) }[/math] algorithm for the inversion of general Toeplitz matrices". Computers & Mathematics with Applications 50 (5–6): 741–752. doi:10.1016/j.camwa.2005.03.011. http://amath.colorado.edu/faculty/martinss/Pubs/2004_toeplitz.pdf. 
• S. Schechter (1959). "On the inversion of certain matrices". Mathematical Tables and Other Aids to Computation 13 (66): 73–77. doi:10.2307/2001955. https://www.ams.org/journals/mcom/1959-13-066/S0025-5718-1959-0105798-2/S0025-5718-1959-0105798-2.pdf. 
• 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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