Cauchy Determinant
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Cauchy Determinant

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 injective sequences (they contain distinct elements).

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

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 and . If the sequences were not injective, the determinant would vanish, and tends to infinity if some tends to . 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

    (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 Lagrange polynomials for and , respectively. That is,

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),
  • (pivoted) LU factorization with 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. 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 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.

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