HilbertMatrix

The (possibly rectangular) Hilbert matrix, entry (i, j) = 1/(i + j - 1), kept exact.

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HilbertMatrix(n)the n×n Hilbert matrix

HilbertMatrix({m, n})the m×n Hilbert matrix

Typed signatures

Type	Package	Applies
(integer | list<integer>) -> list<any>	combinatorics	always

Domain: Collections

Details

• Every entry is an exact rational, never a float — Wolfram's own default.
• Symmetric when square, and famously ill-conditioned (its determinant shrinks fast: det(HilbertMatrix(3)) = 1/2160).
• The rectangular dimensions are a {m, n} LIST, not two bare arguments — kernel-checked: Wolfram's HilbertMatrix has no 2-argument form, only HilbertMatrix[n] and HilbertMatrix[{m, n}].

Examples

Basic #

live ↗

In$\mathrm{HilbertMatrix}(3)$Out$\bigl\lbrack\bigl\lbrack1, \frac{1}{2}, \frac{1}{3}\bigr\rbrack, \bigl\lbrack\frac{1}{2}, \frac{1}{3}, \frac{1}{4}\bigr\rbrack, \bigl\lbrack\frac{1}{3}, \frac{1}{4}, \frac{1}{5}\bigr\rbrack\bigr\rbrack$

Scope #

live ↗

In$\mathrm{HilbertMatrix}(\bigl\lbrack2, 3\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack1, \frac{1}{2}, \frac{1}{3}\bigr\rbrack, \bigl\lbrack\frac{1}{2}, \frac{1}{3}, \frac{1}{4}\bigr\rbrack\bigr\rbrack$

About

Examples	2
Checked against	Wolfram Language 2 agree
Crosswalk	FungrimWolfram Language

See also: DiagonalMatrix