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linalg::hilbert(n) returns the n×n Hilbert
matrix H = (hi, j)1
≤ i ≤ m, 1 ≤ j ≤ n defined
The entries of Hilbert matrices are rational numbers. Note,
however, that the returned matrix is not defined over the component
but over the standard component domain
Thus, no conversion is necessary when working with other functions
that expect or return matrices over that component domain.
linalg::hilbert(n, Dom::Rational) to
define the n×n Hilbert
matrix over the field of rational numbers.
We construct the 3×3 Hilbert matrix:
H := linalg::hilbert(3)
This is a matrix of the domain
If you prefer a different component ring, the matrix may be
converted to the desired domain after construction (see
coerce, for example).
Alternatively, one can specify the component ring when creating the
Hilbert matrix. For example, specification of the domain
Dom::Float generates floating-point
H := linalg::hilbert(3, Dom::Float)
domtype( H )
The dimension of the matrix: a positive integer
of the domain
Hilbert matrices are symmetric and positive definite.
Hilbert matrices of large dimension are notoriously ill-conditioned
challenging any numerical inversion scheme. However, their inverse
can also be computed by a closed formula (see