In
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrice ...
, a Hilbert matrix, introduced by , is a
square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
with entries being the
unit fraction
A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/''n''. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc ...
s
:
For example, this is the 5 × 5 Hilbert matrix:
:
The Hilbert matrix can be regarded as derived from the integral
:
that is, as a
Gramian matrix for powers of ''x''. It arises in the
least squares
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the re ...
approximation of arbitrary functions by
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s.
The Hilbert matrices are canonical examples of
ill-conditioned
In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive a function is to changes or errors in the input ...
matrices, being notoriously difficult to use in
numerical computation
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
. For example, the 2-norm
condition number
In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive a function is to changes or errors in the inpu ...
of the matrix above is about 4.8.
Historical note
introduced the Hilbert matrix to study the following question in
approximation theory
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. Note that what is meant by ''best'' and ''simpler'' wil ...
: "Assume that , is a real interval. Is it then possible to find a non-zero polynomial ''P'' with integer coefficients, such that the integral
:
is smaller than any given bound ''ε'' > 0, taken arbitrarily small?" To answer this question, Hilbert derives an exact formula for the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the Hilbert matrices and investigates their asymptotics. He concludes that the answer to his question is positive if the length of the interval is smaller than 4.
Properties
The Hilbert matrix is
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
and
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite fu ...
. The Hilbert matrix is also
totally positive (meaning that the determinant of every
submatrix
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object.
For example,
\begi ...
is positive).
The Hilbert matrix is an example of a
Hankel matrix. It is also a specific example of a
Cauchy matrix.
The determinant can be expressed in
closed form, as a special case of the
Cauchy determinant. The determinant of the ''n'' × ''n'' Hilbert matrix is
:
where
:
Hilbert already mentioned the curious fact that the determinant of the Hilbert matrix is the reciprocal of an integer (see sequence in the
OEIS
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...
), which also follows from the identity
:
Using
Stirling's approximation
In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less p ...
of the
factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) \ ...
, one can establish the following asymptotic result:
:
where ''a''
''n'' converges to the constant
as
, where ''A'' is the
Glaisher–Kinkelin constant.
The
inverse
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when a ...
of the Hilbert matrix can be expressed in closed form using
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s; its entries are
:
where ''n'' is the order of the matrix.
It follows that the entries of the inverse matrix are all integers, and that the signs form a checkerboard pattern, being positive on the
principal diagonal
In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A is the list of entries a_ where i = j. All off-diagonal elements are zero in a diagonal matrix. ...
. For example,
:
The condition number of the ''n'' × ''n'' Hilbert matrix grows as
.
Applications
The Method of moments (statistics), method of moments applied to polynomial distributions results in a
Hankel matrix, which in the special case of approximating a probability distribution on the interval
, 1results in a Hilbert matrix. This matrix needs to be inverted to obtain the weight parameters of the polynomial distribution approximation.
[J. Munkhammar, L. Mattsson, J. Rydén (2017]
"Polynomial probability distribution estimation using the method of moments"
PLoS ONE 12(4): e0174573.
References
Further reading
*. Reprinted in
*
*
*
*
{{Matrix classes
Numerical linear algebra
Approximation theory
Matrices
Determinants