HilbertMatrix
gives the n×n Hilbert matrix with elements of the form .
HilbertMatrix[{m,n}]
gives the m×n Hilbert matrix.
Details and Options

- HilbertMatrix[n] or HilbertMatrix[{m,n}] gives a matrix with exact rational entries.
- HilbertMatrix[…,WorkingPrecision->p] gives a matrix with entries of precision p.
Examples
open allclose allScope (2)
Options (1)
Applications (2)
Find the exact inverse of the 3×3 Hilbert matrix:
Hilbert matrices are often used to compare numerical algorithms:
Compare methods for solving for known
:

Solve using LinearSolve with Gaussian elimination:

Solve using LinearSolve using a Cholesky decomposition:
Solve using LeastSquares:
Properties & Relations (5)
Square Hilbert matrices are real symmetric and positive definite:
Hilbert matrices can be expressed in terms of HankelMatrix:
Compare with HilbertMatrix:
Hilbert matrices can be expressed in terms of CauchyMatrix:
Compare with HilbertMatrix:
The smallest eigenvalue of a square Hilbert matrix decreases exponentially with n:
The model is a reasonable predictor of magnitude for larger values of n:
The condition number increases exponentially with n:
The 2-norm condition number is the ratio of largest to smallest eigenvalue due to symmetry:
Neat Examples (3)
Text
Wolfram Research (2007), HilbertMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/HilbertMatrix.html.
CMS
Wolfram Language. 2007. "HilbertMatrix." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HilbertMatrix.html.
APA
Wolfram Language. (2007). HilbertMatrix. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HilbertMatrix.html