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| In [[linear algebra]], a '''Hilbert matrix''', introduced by {{harvs|txt|last=Hilbert|year=1894|authorlink=David Hilbert}}, is a [[square matrix]] with entries being the [[unit fraction]]s
| | My name: Salvatore Coverdale<br>My age: 34 years old<br>Country: United States<br>Home town: Washington <br>Postal code: 20005<br>Street: 1748 Forest Drive |
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| :<math> H_{ij} = \frac{1}{i+j-1}. </math> | |
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| For example, this is the 5 × 5 Hilbert matrix:
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| :<math>H = \begin{bmatrix} | |
| 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\[4pt]
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| \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} \\[4pt]
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| \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} \\[4pt]
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| \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} \\[4pt]
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| \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9} \end{bmatrix}.</math>
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| The Hilbert matrix can be regarded as derived from the integral
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| :<math> H_{ij} = \int_{0}^{1} x^{i+j} \, dx, </math> | |
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| that is, as a [[Gramian matrix]] for powers of ''x''. It arises in the [[least squares]] approximation of arbitrary functions by [[polynomial]]s.
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| The Hilbert matrices are canonical examples of [[ill-conditioned]] matrices, making them notoriously difficult to use in numerical computation. For example, the 2-norm [[condition number]] of the matrix above is about 4.8 · 10<sup>5</sup>.
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| ==Historical note==
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| {{harvtxt|Hilbert|1894}} introduced the Hilbert matrix to study the following question in [[approximation theory]]: "Assume that {{nowrap|''I'' {{=}} [''a'', ''b'']}} is a real interval. Is it then possible to find a non-zero polynomial ''P'' with integral coefficients, such that the integral
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| :<math>\int_{a}^b P(x)^2 dx</math> | |
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| is smaller than any given bound ''ε'' > 0, taken arbitrarily small?" To answer this question, Hilbert derives an exact formula for the [[determinant]] of the Hilbert matrices and investigates their asymptotics. He concludes that the answer to his question is positive if the length {{nowrap|''b'' − ''a''}} of the interval is smaller than 4.
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| ==Properties==
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| The Hilbert matrix is [[symmetric matrix|symmetric]] and [[positive-definite matrix|positive definite]]. The Hilbert matrix is also [[totally positive]] (meaning the determinant of every [[submatrix]] is positive).
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| The Hilbert matrix is an example of a [[Hankel matrix]].
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| The determinant can be expressed in [[closed-form expression|closed form]], as a special case of the [[Cauchy determinant]]. The determinant of the ''n'' × ''n'' Hilbert matrix is
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| :<math>\det(H)={{c_n^{\;4}}\over {c_{2n}}}</math> | |
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| where
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| :<math>c_n = \prod_{i=1}^{n-1} i^{n-i}=\prod_{i=1}^{n-1} i!.\,</math>
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| Hilbert already mentioned the curious fact that the determinant of the Hilbert matrix is the reciprocal of an integer (see sequence {{OEIS2C|A005249}} in the [[OEIS]]) which also follows from the identity
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| : <math>{1 \over \det (H)}={{c_{2n}}\over {c_n^{\;4}}}=n!\cdot \prod_{i=1}^{2n-1} {i \choose [i/2]}.
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| </math>
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| Using [[Stirling's approximation]] of the [[factorial]] one can establish the following asymptotic result:
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| :<math>\det(H)=a_n\, n^{-1/4}(2\pi)^n \,4^{-n^2}</math>
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| where ''a''<sub>''n''</sub> converges to the constant <math>e^{1/4} 2^{1/12} A^{ - 3} \approx 0.6450 </math> as <math>n\rightarrow\infty</math>, where A is the [[Glaisher-Kinkelin constant]].
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| The [[matrix inverse|inverse]] of the Hilbert matrix can be expressed in closed form using [[binomial coefficient]]s; its entries are
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| :<math>(H^{-1})_{ij}=(-1)^{i+j}(i+j-1){n+i-1 \choose n-j}{n+j-1 \choose n-i}{i+j-2 \choose i-1}^2</math>
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| where ''n'' is the order of the matrix. It follows that the entries of the inverse matrix are all integer.
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| The condition number of the ''n''-by-''n'' Hilbert matrix grows as <math>O((1+\sqrt{2})^{4n}/\sqrt{n})</math>.
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| ==References==
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| *{{Citation | last1=Hilbert | first1=David | author1-link=David Hilbert | title=Ein Beitrag zur Theorie des Legendre'schen Polynoms | publisher=Springer Netherlands | doi=10.1007/BF02418278 | jfm=25.0817.02 | year=1894 | journal=[[Acta Mathematica]] | issn=0001-5962 | volume=18 | pages=155–159}}. Reprinted in {{cite book|first=David|last= Hilbert|title=Collected papers|volume= II|chapter= article 21}}
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| * {{cite journal|doi=10.1007/PL00005392|last=Beckermann|first=Bernhard|title=The condition number of real Vandermonde, Krylov and positive definite Hankel matrices|journal= Numerische Mathematik|volume=85|issue=4|pages= 553–577|year= 2000}}
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| * {{cite journal|doi=10.2307/2975779|last=Choi|first= M.-D.|title= Tricks or Treats with the Hilbert Matrix|journal=American Mathematical Monthly|volume=90|issue=5|pages=301–312|year= 1983|jstor=2975779}}
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| * {{cite journal|last=Todd|first= John|title=The Condition Number of the Finite Segment of the Hilbert Matrix|journal=National Bureau of Standards, Applied Mathematics Series|volume=39|pages= 109–116|year=1954}}
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| * {{Cite book|last=Wilf|first=H. S.|title=Finite Sections of Some Classical Inequalities|location= Heidelberg|publisher= Springer|year= 1970|isbn=3-540-04809-X}}
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| {{Numerical linear algebra}}
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| [[Category:Numerical linear algebra]]
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| [[Category:Approximation theory]]
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| [[Category:Matrices]]
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| [[Category:Determinants]]
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My name: Salvatore Coverdale
My age: 34 years old
Country: United States
Home town: Washington
Postal code: 20005
Street: 1748 Forest Drive