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Hilbert matrix

From Wikipedia, the free encyclopedia
Square matrix where a[i,j]=1/(i+j-1)

Inlinear algebra, aHilbert matrix, introduced byHilbert (1894), is asquare matrix with entries being theunit fractions

Hij=1i+j−1.{\displaystyle H_{ij}={\frac {1}{i+j-1}}.}

For example, this is the 5 × 5 Hilbert matrix:

H=[1121314151213141516131415161714151617181516171819].{\displaystyle H={\begin{bmatrix}1&{\frac {1}{2}}&{\frac {1}{3}}&{\frac {1}{4}}&{\frac {1}{5}}\\{\frac {1}{2}}&{\frac {1}{3}}&{\frac {1}{4}}&{\frac {1}{5}}&{\frac {1}{6}}\\{\frac {1}{3}}&{\frac {1}{4}}&{\frac {1}{5}}&{\frac {1}{6}}&{\frac {1}{7}}\\{\frac {1}{4}}&{\frac {1}{5}}&{\frac {1}{6}}&{\frac {1}{7}}&{\frac {1}{8}}\\{\frac {1}{5}}&{\frac {1}{6}}&{\frac {1}{7}}&{\frac {1}{8}}&{\frac {1}{9}}\end{bmatrix}}.}

The entries can also be defined by the integral

Hij=∫01xi+j−2dx,{\displaystyle H_{ij}=\int _{0}^{1}x^{i+j-2}\,dx,}

that is, as aGramian matrix for powers ofx. It arises in theleast squares approximation of arbitrary functions bypolynomials.

The Hilbert matrices are canonical examples ofill-conditioned matrices, being notoriously difficult to use innumerical computation. For example, the 2-normcondition number of the matrix above is about 4.8×105.

Historical note

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Hilbert (1894) introduced the Hilbert matrix to study the following question inapproximation theory: "Assume thatI = [a,b], is a real interval. Is it then possible to find a non-zero polynomialP with integer coefficients, such that the integral

∫abP(x)2dx{\displaystyle \int _{a}^{b}P(x)^{2}dx}

is smaller than any given boundε > 0, taken arbitrarily small?" To answer this question, Hilbert derives an exact formula for thedeterminant of the Hilbert matrices and investigates their asymptotics. He concludes that the answer to his question is positive if the lengthb −a of the interval is smaller than 4.

Properties

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The Hilbert matrix issymmetric andpositive definite. The Hilbert matrix is alsototally positive (meaning that the determinant of everysubmatrix is positive).

The Hilbert matrix is an example of aHankel matrix. It is also a specific example of aCauchy matrix.

The determinant can be expressed inclosed form, as a special case of theCauchy determinant. The determinant of then ×n Hilbert matrix is

det(H)=cn4c2n,{\displaystyle \det(H)={\frac {c_{n}^{4}}{c_{2n}}},}

where

cn=∏i=1n−1in−i=∏i=1n−1i!.{\displaystyle c_{n}=\prod _{i=1}^{n-1}i^{n-i}=\prod _{i=1}^{n-1}i!.}

Hilbert already mentioned the curious fact that the determinant of the Hilbert matrix is the reciprocal of an integer (see sequenceOEIS: A005249 in theOEIS), which also follows from the identity

1det(H)=c2ncn4=n!⋅∏i=12n−1(i[i/2]).{\displaystyle {\frac {1}{\det(H)}}={\frac {c_{2n}}{c_{n}^{4}}}=n!\cdot \prod _{i=1}^{2n-1}{\binom {i}{[i/2]}}.}

UsingStirling's approximation of thefactorial, one can establish the following asymptotic result:

det(H)∼ann−1/4(2π)n4−n2,{\displaystyle \det(H)\sim a_{n}\,n^{-1/4}(2\pi )^{n}\,4^{-n^{2}},}

wherean converges to the constante1/421/12A−3≈0.6450{\displaystyle e^{1/4}\,2^{1/12}\,A^{-3}\approx 0.6450} asn→∞{\displaystyle n\to \infty }, whereA is theGlaisher–Kinkelin constant.

Theinverse of the Hilbert matrix can be expressed in closed form usingbinomial coefficients; its entries are

(H−1)ij=(−1)i+j(i+j−1)(n+i−1n−j)(n+j−1n−i)(i+j−2i−1)2,{\displaystyle (H^{-1})_{ij}=(-1)^{i+j}(i+j-1){\binom {n+i-1}{n-j}}{\binom {n+j-1}{n-i}}{\binom {i+j-2}{i-1}}^{2},}

wheren is the order of the matrix.[1] It follows that the entries of the inverse matrix are all integers, and that the signs form a checkerboard pattern, being positive on theprincipal diagonal. For example,

[1121314151213141516131415161714151617181516171819]−1=[25−3001050−1400630−3004800−1890026880−126001050−1890079380−11760056700−140026880−117600179200−88200630−1260056700−8820044100].{\displaystyle {\begin{bmatrix}1&{\frac {1}{2}}&{\frac {1}{3}}&{\frac {1}{4}}&{\frac {1}{5}}\\{\frac {1}{2}}&{\frac {1}{3}}&{\frac {1}{4}}&{\frac {1}{5}}&{\frac {1}{6}}\\{\frac {1}{3}}&{\frac {1}{4}}&{\frac {1}{5}}&{\frac {1}{6}}&{\frac {1}{7}}\\{\frac {1}{4}}&{\frac {1}{5}}&{\frac {1}{6}}&{\frac {1}{7}}&{\frac {1}{8}}\\{\frac {1}{5}}&{\frac {1}{6}}&{\frac {1}{7}}&{\frac {1}{8}}&{\frac {1}{9}}\end{bmatrix}}^{-1}=\left[{\begin{array}{rrrrr}25&-300&1050&-1400&630\\-300&4800&-18900&26880&-12600\\1050&-18900&79380&-117600&56700\\-1400&26880&-117600&179200&-88200\\630&-12600&56700&-88200&44100\end{array}}\right].}

The condition number of then × n Hilbert matrix grows asO((1+2)4n/n){\displaystyle O\left(\left(1+{\sqrt {2}}\right)^{4n}/{\sqrt {n}}\right)}.

Applications

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Themethod of moments applied to polynomial distributions results in aHankel matrix, which in the special case of approximating a probability distribution on the interval [0, 1] results in a Hilbert matrix. This matrix needs to be inverted to obtain the weight parameters of the polynomial distribution approximation.[2]

References

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  1. ^Choi, Man-Duen (1983). "Tricks or Treats with the Hilbert Matrix".The American Mathematical Monthly.90 (5):301–312.doi:10.2307/2975779.JSTOR 2975779.
  2. ^Munkhammar, Joakim; Mattsson, Lars; Rydén, Jesper (2017)."Polynomial probability distribution estimation using the method of moments".PLOS ONE.12 (4) e0174573.Bibcode:2017PLoSO..1274573M.doi:10.1371/journal.pone.0174573.PMC 5386244.PMID 28394949.

Further reading

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Matrix classes
Explicitly constrained entries
Constant
Conditions oneigenvalues or eigenvectors
Satisfying conditions onproducts orinverses
With specific applications
Used instatistics
Used ingraph theory
Used in science and engineering
Related terms
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