Properties
The Hilbert matrix is symmetric and positive definite. The Hilbert matrix is also totally positive (meaning the determinant of every submatrix is positive).
The Hilbert matrix is an example of a Hankel matrix.
The determinant can be expressed in closed form, as a special case of the Cauchy determinant. The determinant of the n × n Hilbert matrix is
where
Hilbert already mentioned the curious fact that the determinant of the Hilbert matrix is the reciprocal of an integer (see sequence A005249 in the OEIS) which also follows from the identity
Using Stirling's approximation of the factorial one can establish the following asymptotic result:
where an converges to the constant as, where A is the Glaisher-Kinkelin constant.
The inverse of the Hilbert matrix can be expressed in closed form using binomial coefficients; its entries are
where n is the order of the matrix. It follows that the entries of the inverse matrix are all integer.
The condition number of the n-by-n Hilbert matrix grows as .
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Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)