notatio

HankelMatrix ​

The square matrix, constant along every anti-diagonal, built from c.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageHankelMatrix✓
HankelMatrix(c)the n×n Hankel matrix with first column and first row c, zero-padded past c's reach
HankelMatrix(c, r)the n×m Hankel matrix (n = Length(c), m = Length(r)) with r as the last row
Typed signatures
TypePackageApplies
(collection<any> | integer, collection<any>?) -> list<list<any>>combinatoricsalways

Domain: Collections

Details
  • A Hankel matrix is constant along each anti-diagonal: M(i, j) depends only on i + j.
  • r's first element coincides with c's last (both are the matrix's shared corner), so it never surfaces on its own.

Examples

Basic #

Past c's reach, the matrix is zero-padded

live ↗
InHankelMatrix([1,2,3])\mathrm{HankelMatrix}(\bigl\lbrack1, 2, 3\bigr\rbrack)Out[[1,2,3],[2,3,0],[3,0,0]]\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack2, 3, 0\bigr\rbrack, \bigl\lbrack3, 0, 0\bigr\rbrack\bigr\rbrack
live ↗
InHankelMatrix(3)\mathrm{HankelMatrix}(3)Out[[1,2,3],[2,3,0],[3,0,0]]\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack2, 3, 0\bigr\rbrack, \bigl\lbrack3, 0, 0\bigr\rbrack\bigr\rbrack
Scope #

A second argument becomes the matrix's last row

live ↗
InHankelMatrix([1,2,3],[3,4,5])\mathrm{HankelMatrix}(\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack3, 4, 5\bigr\rbrack)Out[[1,2,3],[2,3,4],[3,4,5]]\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack2, 3, 4\bigr\rbrack, \bigl\lbrack3, 4, 5\bigr\rbrack\bigr\rbrack
live ↗
InHankelMatrix([[17,15,13,1],[15,13,1,−1],[13,1,−1,−13],[1,−1,−13,−15]])\mathrm{HankelMatrix}(\bigl\lbrack\bigl\lbrack\frac{1}{7}, \frac{1}{5}, \frac{1}{3}, 1\bigr\rbrack, \bigl\lbrack\frac{1}{5}, \frac{1}{3}, 1, -1\bigr\rbrack, \bigl\lbrack\frac{1}{3}, 1, -1, -\frac{1}{3}\bigr\rbrack, \bigl\lbrack1, -1, -\frac{1}{3}, -\frac{1}{5}\bigr\rbrack\bigr\rbrack)Out[[[17,15,13,1],[15,13,1,−1],[13,1,−1,−13],[1,−1,−13,−15]],[[15,13,1,−1],[13,1,−1,−13],[1,−1,−13,−15],0],[[13,1,−1,−13],[1,−1,−13,−15],0,0],[[1,−1,−13,−15],0,0,0]]\bigl\lbrack\bigl\lbrack\bigl\lbrack\frac{1}{7}, \frac{1}{5}, \frac{1}{3}, 1\bigr\rbrack, \bigl\lbrack\frac{1}{5}, \frac{1}{3}, 1, -1\bigr\rbrack, \bigl\lbrack\frac{1}{3}, 1, -1, -\frac{1}{3}\bigr\rbrack, \bigl\lbrack1, -1, -\frac{1}{3}, -\frac{1}{5}\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\frac{1}{5}, \frac{1}{3}, 1, -1\bigr\rbrack, \bigl\lbrack\frac{1}{3}, 1, -1, -\frac{1}{3}\bigr\rbrack, \bigl\lbrack1, -1, -\frac{1}{3}, -\frac{1}{5}\bigr\rbrack, 0\bigr\rbrack, \bigl\lbrack\bigl\lbrack\frac{1}{3}, 1, -1, -\frac{1}{3}\bigr\rbrack, \bigl\lbrack1, -1, -\frac{1}{3}, -\frac{1}{5}\bigr\rbrack, 0, 0\bigr\rbrack, \bigl\lbrack\bigl\lbrack1, -1, -\frac{1}{3}, -\frac{1}{5}\bigr\rbrack, 0, 0, 0\bigr\rbrack\bigr\rbrack
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Examples4
Checked againstWolfram Language 3 agree1 unevaluated
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