A square matrix with constant skew diagonals. In other words, a Hankel matrix is a matrix in which the 
th entry depends only on the sum 
. 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].](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FHankelMatrix%2FNumberedEquation1.svg) |
(1) |
An upper triangular Hankel matrix with first column and row 
can be specified in the Wolfram
Language as HankelMatrix[
c1, ..., cn
], and HankelMatrix[n]
where 
is an integer gives the 
matrix 
with first row and column equal to 
and with every element below the main skew
diagonal equal to 0. The first few matrices 
are given by
The elements of this Hankel matrix are given explicitly by
 |
(5) |
The determinant of 
is given by 
, where 
is the floor function,
so the first few values are 1, 
, 
,
256, 3125, 
,
, 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."
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Hankel Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HankelMatrix.html