Signature matrix - Revision history

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2014-12-09T14:24:01Z

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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~~		Hi there, I am Alyson Boon even though it is not the name on my beginning certificate. Distributing manufacturing is where my main earnings comes from and it's some thing I really appreciate. It's not a typical thing but what she likes doing is to perform domino but she doesn't have [http://myfusionprofits.com/groups/find-out-about-self-improvement-tips-here/ free psychic readings] the time lately. My wife and I live [http://c045.danah.co.kr/home/index.php?document_srl=1356970&mid=qna live psychic reading] in Mississippi and I love every working day living here.<br><br>Also visit my blog ... real psychic ([http://www.chk.woobi.co.kr/xe/?document_srl=346069 www.chk.woobi.co.kr])

	~~:<math>~~
	~~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>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~~'~~').~~

	~~The~~ [~~[Hilbert matrix]] is a special case of the Cauchy matrix, where~~
	:~~<math>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>(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:~~

	~~The determinant of a square Cauchy matrix '''A''' is known as a '''Cauchy determinant''' and can be given explicitly as~~
	~~:<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> &emsp;&emsp;&emsp;&emsp;(Schechter 1959, eqn 4)~~.
	~~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~~
	:~~<math>b_{ij} = (x_j - y_i) A_j(y_i) B_i(x_j) \,<~~/~~math> &emsp;&emsp;&emsp;&emsp;(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,~~
	~~:<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>~~
	~~with~~
	~~:<math>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>C_{ij}=\frac{r_i s_j}{x_i-y_j}~~.</~~math>~~

	~~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]]~~

	~~:<math>\mathbf{XC}-\mathbf{CY}=rs^\mathrm{T}</math>~~

	~~(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~~
	* approximate Cauchy matrix-vector multiplication with <~~math~~>~~O(n \log n)~~<~~/math~~> ~~[[FLOPS\| ops]] (e~~.g. ~~the [[fast multipole method]]),~~
	* ([[pivot element\|pivoted]]) [[LU factorization]] with <math>O(n^2)</math> ops (GKO algorithm), and thus linear system solving,
	* approximated or unstable algorithms for linear system solving in <math>O(n \log^2 n)</math>.
	~~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).~~

	~~==See also==~~
	*[~~[Toeplitz matrix]]~~

	~~==References==~~
	* {{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}}~~
	* {{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}}~~
	* {{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}}
	* {{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}}

	~~[[Category:Matrices]]~~
	~~[[Category:Determinants]]~~

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Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q4921154

← Older revision		Revision as of 10:36, 16 March 2013
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	~~Wilber Berryhill~~ is the ~~title his mothers~~ and ~~fathers gave him~~ and ~~he completely digs~~ that ~~title~~. ~~Her family life in Ohio~~. ~~My day job~~ is ~~an invoicing officer but I~~'~~ve already utilized~~ for ~~an additional~~ 1. ~~To play lacross~~ is ~~something he would by no means give up~~.<br><br>~~Take~~ a ~~look at my blog~~ - [http://www.~~khuplaza~~.~~com~~/~~dent~~/~~14869889 psychic phone~~]		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

			:<math>
			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>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'').

			The [[Hilbert matrix]] is a special case of the Cauchy matrix, where
			:<math>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>(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:

			The determinant of a square Cauchy matrix '''A''' is known as a '''Cauchy determinant''' and can be given explicitly as
			:<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> &emsp;&emsp;&emsp;&emsp;(Schechter 1959, eqn 4).
			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
			:<math>b_{ij} = (x_j - y_i) A_j(y_i) B_i(x_j) \,</math> &emsp;&emsp;&emsp;&emsp;(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,
			:<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>
			with
			:<math>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>C_{ij}=\frac{r_i s_j}{x_i-y_j}.</math>

			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]]

			:<math>\mathbf{XC}-\mathbf{CY}=rs^\mathrm{T}</math>

			(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
			* approximate Cauchy matrix-vector multiplication with <math>O(n \log n)</math> [[FLOPS\| ops]] (e.g. the [[fast multipole method]]),
			* ([[pivot element\|pivoted]]) [[LU factorization]] with <math>O(n^2)</math> ops (GKO algorithm), and thus linear system solving,
			* approximated or unstable algorithms for linear system solving in <math>O(n \log^2 n)</math>.
			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).

			==See also==
			*[[Toeplitz matrix]]

			==References==
			* {{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}}
			* {{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}}
			* {{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}}
			* {{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}}

			[[Category:Matrices]]
			[[Category:Determinants]]

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2012-01-01T15:10:33Z

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