TOPICS
Search

Hankel Matrix


A square matrix with constant skew diagonals. In other words, a Hankel matrix is a matrix in which the (i,j)th entry depends only on the sum i+j. Such matrices are sometimes known as persymmetric matrices or, in older literature, orthosymmetric matrices.

In the Wolfram Language, such a Hankel matrix can be generated for example by HankelMatrix[{a, b, c, d, e}, {e, f, g, h, i}], giving

 [a b c d e; b c d e f; c d e f g; d e f g h; e f g h i].
(1)

An upper triangular Hankel matrix with first column and row c can be specified in the Wolfram Language as HankelMatrix[{c1, ..., cn}], and HankelMatrix[n] where n>0 is an integer gives the n×n matrix H_n with first row and column equal to {1,2,...,n} and with every element below the main skew diagonal equal to 0. The first few matrices H_n are given by

H_2=[1 2; 2 0]
(2)
H_3=[1 2 3; 2 3 0; 3 0 0]
(3)
H_4=[1 2 3 4; 2 3 4 0; 3 4 0 0; 4 0 0 0].
(4)

The elements of this Hankel matrix are given explicitly by

 h_(ij)={0   if i+j-1>n; i+j-1   otherwise.
(5)

The determinant of H_n is given by det(H_n)=(-1)^(|_n/2_|)n^n, where |_n_| is the floor function, so the first few values are 1, -4, -27, 256, 3125, -46656, -823543, 16777216, ... (OEIS A000312).


See also

Antisymmetric Matrix, Diagonal Matrix, Skew Diagonal, Symmetric Matrix, Triangular Matrix

Explore with Wolfram|Alpha

References

Mays, M. E. and Wojciechowski, J. "A Determinant Property of Catalan Numbers." Disc. Math. 211, 125-133, 2000.Sloane, N. J. A. Sequence A000312/M3619 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "Hankel Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HankelMatrix.html

Subject classifications