Cauchy matrix
In mathematics, a Cauchy matrix, named after Augustin-Louis Cauchy, is an m×n matrix with elements aij in the form
where and are elements of a field , and and are
The Hilbert matrix is a special case of the Cauchy matrix, where
Every
Cauchy determinants
The determinant of a Cauchy matrix is clearly a
The determinant of a square Cauchy matrix A is known as a Cauchy determinant and can be given explicitly as
- (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 = [bij] is given by
- (Schechter 1959, Theorem 1)
where Ai(x) and Bi(x) are the
with
Generalization
A matrix C is called Cauchy-like if it is of the form
Defining X=diag(xi), Y=diag(yi), one sees that both Cauchy and Cauchy-like matrices satisfy the displacement equation
(with 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 ops (e.g. the fast multipole method),
- (LU factorizationwith ops (GKO algorithm), and thus linear system solving,
- approximated or unstable algorithms for linear system solving in .
Here denotes the size of the matrix (one usually deals with square matrices, though all algorithms can be easily generalized to rectangular matrices).
See also
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. 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. .
- P. G. Martinsson; M. Tygert; V. Rokhlin (2005). "An algorithm for the inversion of general Toeplitz matrices" (PDF). Computers & Mathematics with Applications. 50 (5–6): 741–752. .
- S. Schechter (1959). "On the inversion of certain matrices" (PDF). Mathematical Tables and Other Aids to Computation. 13 (66): 73–77. 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 .