Hilbert-Matrix

The Hilbert matrix of order is the following square, symmetric, positive definite matrix:

The individual components are therefore given by. The historical approach corresponds to the representation with integral:.

It is defined by German mathematicians David Hilbert in 1894 in conjunction with the theory of the Legendre polynomials. Since the matrix is positive definite, there exists its inverse, that is, a system of linear equations with this coefficient is uniquely solvable. The Hilbert matrix or the system of equations in question is comparatively ill-conditioned, ie, the worse is the greater. The condition number grows exponentially with; the condition number of is 526,16 ( Frobenius norm ), that of 15'613, 8 That is, ( the resolution of the equation system ) is always larger numbers occur, the greater the calculation of the inverse. Therefore, the Hilbert matrix is a classic test case for computer programs for the inversion of matrices and solution of linear systems of equations, such as the Gauss method, LU decomposition, Cholesky decomposition, etc. All the components of the inverse matrix are all numbers with alternating signs.

The components of the inverse of the Hilbert matrix can be calculated directly by closed formulas:

Which can be expressed by the binomial coefficients:

In the special case reduces to:

That the inverse of the Hilbert matrix can be calculated exactly, is particularly useful when, for example, in a test, the result of the numerical inversion of a matrix with a Hilbert - LR or Cholesky decomposition, which is of course affected by rounding errors judged should be.

Determinant

The determinant of the inverse of the Hilbert matrix can also be calculated accurately by means of the following formula:

As a determinant of the Hilbert matrix thus obtained with the reciprocal of the inverse. The determinants of the inverse for loud so that 1, 12, 2160, 6.048 million and 266,716,800,000th

Numerical examples for inverse

From the above formulas obtained for the (exact ) inverse in the cases:

For their own experiments with Hilbert (and of course with all other ) matrices modern mathematics software packages like MATLAB, Maple, Mathematica or GNU Octave useful. For example with Mathematica the last inverse can be calculated using the following command:

Inverse to calculate:

In: = Inverse [ Hilbert matrix ] / / TraditionalForm The ill-conditioning of the Hilbert matrix practically means that the row (and consequently the column ) vectors are nearly linearly dependent. Geometrically, this manifests itself, inter alia, that the angles between the row vectors are very small, between the last row vectors each smallest; Thus, for example, the angle between the last and the penultimate row vector of less than 3 ° (in radians: less than). For larger angles are correspondingly smaller. The angle between the first line vector of the plane and which is spanned by the other two row vectors is a little smaller than 1.3 °, the corresponding angles for the other two row vectors is still smaller; these angles are even smaller for larger.