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| In [[mathematics]], a '''Cauchy matrix''', named after [[Augustin Louis Cauchy]], is an ''m''×''n'' [[matrix (mathematics)|matrix]] with elements ''a''<sub>''ij''</sub> in the form
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| :<math>
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| 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
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| </math>
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| where <math>x_i</math> and <math>y_j</math> are elements of a [[field (mathematics)|field]] <math>\mathcal{F}</math>, and <math>(x_i)</math> and <math>(y_j)</math> are [[injective]] sequences (they do not contain repeated elements; elements are ''distinct''). | |
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| The [[Hilbert matrix]] is a special case of the Cauchy matrix, where
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| :<math>x_i-y_j = i+j-1. \;</math> | |
| Every [[submatrix]] of a Cauchy matrix is itself a Cauchy matrix.
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| == Cauchy determinants ==
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| The determinant of a Cauchy matrix is clearly a [[rational fraction]] in the parameters <math>(x_i)</math> and <math>(y_j)</math>. If the sequences were not injective, the determinant would vanish, and tends to infinity if some <math>x_i</math> tends to <math>y_j</math>. A subset of its zeros and poles are thus known. The fact is that there are no more zeros and poles:
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| The determinant of a square Cauchy matrix '''A''' is known as a '''Cauchy determinant''' and can be given explicitly as
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| :<math> \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).
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| It is always nonzero, and thus all square Cauchy matrices are [[invertible matrix|invertible]]. The inverse '''A'''<sup>−1</sup> = '''B''' = [b<sub>ij</sub>] is given by
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| :<math>b_{ij} = (x_j - y_i) A_j(y_i) B_i(x_j) \,</math>     (Schechter 1959, Theorem 1) | |
| where ''A''<sub>i</sub>(x) and ''B''<sub>i</sub>(x) are the [[Lagrange polynomials]] for <math>(x_i)</math> and <math>(y_j)</math>, respectively. That is,
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| :<math>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>
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| with
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| :<math>A(x) = \prod_{i=1}^n (x-x_i) \quad\text{and}\quad B(x) = \prod_{i=1}^n (x-y_i). </math>
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| ==Generalization==
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| A matrix '''C''' is called '''Cauchy-like''' if it is of the form
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| :<math>C_{ij}=\frac{r_i s_j}{x_i-y_j}.</math>
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| Defining '''X'''=diag(x<sub>i</sub>), '''Y'''=diag(y<sub>i</sub>), one sees that both Cauchy and Cauchy-like matrices satisfy the [[displacement rank|displacement equation]]
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| :<math>\mathbf{XC}-\mathbf{CY}=rs^\mathrm{T}</math>
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| (with <math>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
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| * approximate Cauchy matrix-vector multiplication with <math>O(n \log n)</math> [[FLOPS| ops]] (e.g. the [[fast multipole method]]),
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| * ([[pivot element|pivoted]]) [[LU factorization]] with <math>O(n^2)</math> ops (GKO algorithm), and thus linear system solving,
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| * approximated or unstable algorithms for linear system solving in <math>O(n \log^2 n)</math>.
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| Here <math>n</math> denotes the size of the matrix (one usually deals with square matrices, though all algorithms can be easily generalized to rectangular matrices).
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| ==See also==
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| *[[Toeplitz matrix]]
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| ==References==
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| * {{cite journal |author=A. Gerasoulis |title=A fast algorithm for the multiplication of generalized Hilbert matrices with vectors |journal=Mathematics of Computation |year=1988 |volume=50 |issue=181 |pages=179–188 |url=http://www.ams.org/journals/mcom/1988-50-181/S0025-5718-1988-0917825-9/S0025-5718-1988-0917825-9.pdf}}
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| * {{cite journal |author=I. Gohberg, T. Kailath, V. Olshevsky |title=Fast Gaussian elimination with partial pivoting for matrices with displacement structure |journal=Mathematics of Computation |year=1995 |volume=64 |issue=212 |pages=1557–1576 |url=http://www.ams.org/journals/mcom/1995-64-212/S0025-5718-1995-1312096-X/S0025-5718-1995-1312096-X.pdf}}
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| * {{cite journal |author=P. G. Martinsson, M. Tygert, V. Rokhlin |title=An <math>O(N \log^2 N)</math> algorithm for the inversion of general Toeplitz matrices |journal=Computers & Mathematics with Applications |year=2005 |volume=50 |pages=741–752 |url=http://amath.colorado.edu/faculty/martinss/Pubs/2004_toeplitz.pdf}}
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| * {{cite journal |author=S. Schechter |title=On the inversion of certain matrices |journal=Mathematical Tables and Other Aids to Computation |year=1959 |volume=13 |issue=66 |pages=73–77 |url=http://www.ams.org/journals/mcom/1959-13-066/S0025-5718-1959-0105798-2/S0025-5718-1959-0105798-2.pdf}}
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| [[Category:Matrices]]
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| [[Category:Determinants]]
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