Creates a Hilbert matrix.

## Usage

hilbert(n)

## Arguments

n

an integer value, the dimension of the square matrix.

## Details

An $$n,n$$ matrix with $$(i,j)$$th element equal to $$1/(i+j-1)$$ is said to be a Hilbert matrix of order $$n$$. Hilbert matrices are symmetric and positive definite.

They are canonical examples of ill-conditioned matrices, making them notoriously difficult to use in numerical computation. For example, the 2-norm condition number of a 5x5 Hilbert matrix above is about 4.8e5.

a matrix

## References

Hilbert D., Collected papers, vol. II, article 21.

Beckermann B, (2000); The condition number of real Vandermonde, Krylov and positive definite Hankel matrices, Numerische Mathematik 85, 553--577, 2000.

Choi, M.D., (1983); Tricks or Treats with the Hilbert Matrix, American Mathematical Monthly 90, 301--312, 1983.

Todd, J., (1954); The Condition Number of the Finite Segment of the Hilbert Matrix, National Bureau of Standards, Applied Mathematics Series 39, 109--116.

Wilf, H.S., (1970); Finite Sections of Some Classical Inequalities, Heidelberg, Springer.

## Examples

## Create a Hilbert Matrix:
H = hilbert(5)
H
#>           [,1]      [,2]      [,3]      [,4]      [,5]
#> [1,] 1.0000000 0.5000000 0.3333333 0.2500000 0.2000000
#> [2,] 0.5000000 0.3333333 0.2500000 0.2000000 0.1666667
#> [3,] 0.3333333 0.2500000 0.2000000 0.1666667 0.1428571
#> [4,] 0.2500000 0.2000000 0.1666667 0.1428571 0.1250000
#> [5,] 0.2000000 0.1666667 0.1428571 0.1250000 0.1111111