A 4×4 Hankel matrix:

A 4×4 Hankel matrix with symbolic entries:

A rectangular Hankel matrix:

Make a Hankel matrix of machine numbers:

Make a Hankel matrix with 20-digit precision:

Hankel matrices with complex entries:

Rectangular Hankel matrices:

A common symbolic notation for Hankel matrices:

Generate a structured Hankel matrix:

The structured representation typically uses much less memory:

HankelMatrix objects include properties that give information about the array:

The "ColumnVector" property gives the first column of the Hankel matrix:

The "RowVector" property gives the last row of the Hankel matrix:

The "Summary" property gives a brief summary of information about the array:

The "StructuredAlgorithms" property lists the functions that have structured algorithms:

When appropriate, structured algorithms return another HankelMatrix object:

The transpose is also a HankelMatrix:

The product of a Hankel matrix and its transpose is no longer a Hankel matrix:

Convert a dense Hankel matrix to a structured Hankel matrix:

An n-term sum of exponential functions:

Sample the exponential function at points:

Use Prony's method [Wikipedia] to recover the sum of exponentials from the data:

The determinant of the Hankel matrix of size is :

A square Hankel matrix with real entries is symmetric:

HankelMatrix[c,RotateRight[c]] is a square anticirculant matrix:

Square anticirculant matrices have eigenvector {1,…} with eigenvalue c₁+c₂+…:

HankelMatrix and ToeplitzMatrix are related by multiplication with an exchange matrix (a reversed identity matrix):

Equivalently, reversing a Hankel matrix gives a Toeplitz matrix:

Demonstrate a factorization of the inverse of a Vandermonde matrix in terms of diagonal, Vandermonde and Hankel matrices:

Express a Cauchy matrix as a product of diagonal, Vandermonde and Hankel matrices:

If element c_m is not the same as r₁, c_m is used and r₁ is ignored:

Visualize the entries of the Hankel matrix: