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HankelMatrix

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

HankelMatrix[n] gives the n x n Hankel matrix with first row and column the integers 1..n.
HankelMatrix[{c1, ..., cm}] gives the m x m Hankel matrix with first column the given list.
HankelMatrix[{c1, ..., cm}, {r1, ..., rn}] gives the m x n Hankel matrix with first column the first list and last row the second.
A Hankel matrix is constant along its antidiagonals; entries are copied verbatim.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= HankelMatrix[4]
Out[1]= {{1, 2, 3, 4}, {2, 3, 4, 0}, {3, 4, 0, 0}, {4, 0, 0, 0}}

In[2]:= HankelMatrix[{a, b, c, d}]
Out[2]= {{a, b, c, d}, {b, c, d, 0}, {c, d, 0, 0}, {d, 0, 0, 0}}

In[3]:= HankelMatrix[{x, y, z}, {z, a, b, c, d}]
Out[3]= {{x, y, z, a, b}, {y, z, a, b, c}, {z, a, b, c, d}}

In[4]:= HankelMatrix[{1, 1 + 2 I, 3 + 4 I}]
Out[4]= {{1, 1 + 2*I, 3 + 4*I}, {1 + 2*I, 3 + 4*I, 0}, {3 + 4*I, 0, 0}}

In[5]:= N[HankelMatrix[3]]
Out[5]= {{1.0, 2.0, 3.0}, {2.0, 3.0, 0.0}, {3.0, 0.0, 0.0}}

Implementation notes

Algorithm. builtin_hankelmatrix builds a matrix constant along antidiagonals, where entry (i,j) depends only on s = i+j-1. Three forms are handled: HankelMatrix[n] (square, antidiagonal index s for s ≤ n else 0, the integer form); HankelMatrix[{c1,…,cm}] (square, first column c, zeros below the antidiagonal); and HankelMatrix[{c…}, {r…}] (m×n, first column c and last row r, with (i,j) = c_s when s ≤ m and r_{s-m} otherwise). The shared corner c_m must equal r_1; if not, it warns via HankelMatrix::crs and uses the column element. Zero arguments emit HankelMatrix::argb; any other shape (non-list, empty list) is left unevaluated.

Data structures. A List of Lists built by hk_build; source entries are deep-copied (expr_copy) so symbolic/complex/exact/inexact entries pass through verbatim — arbitrary precision comes from the entries themselves. Complexity O(mn).

  • Protected.
  • Entries are copied verbatim, so symbolic, exact, complex, machine and

Attributes: Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= HankelMatrix[3]
Out[1]= {{1, 2, 3}, {2, 3, 0}, {3, 0, 0}}

Give an explicit first column; the matrix is constant along every antidiagonal and zero-padded below the secondary diagonal:

In[1]:= HankelMatrix[{a, b, c}]
Out[1]= {{a, b, c}, {b, c, 0}, {c, 0, 0}}

A first column plus a last row builds a rectangular catalecticant — here the shared corner is taken from the column:

In[1]:= HankelMatrix[{1, 2, 3, 4}, {4, 5, 6}]
Out[1]= {{1, 2, 3}, {2, 3, 4}, {3, 4, 5}, {4, 5, 6}}

Hankel determinants encode sequence properties. The catalecticant of the Fibonacci numbers is a perfect power of 2:

In[1]:= Det[HankelMatrix[{1, 1, 2, 3, 5, 8}]]
Out[1]= -262144

Notes

A Hankel matrix is constant along its antidiagonals. With a single integer n, the first row and column are the integers 1..n and the lower-right triangle is zero-filled. Supplying a first column (and optionally a last row) lets you build the catalecticant of any sequence; Det[HankelMatrix[...]] then gives that sequence's Hankel determinant.