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:

The order Padé approximant can computed using Hankel determinants. Generate the first 2n+1 partial sums of the power series for Exp[z]:

Take the second differences of the partial sums:

The Shanks transformation expresses the order Padé approximant as a ratio of two Hankel determinants:

Compare with the result of PadeApproximant:

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:

Determinants of Hankel matrices appear in the continued fraction expansion of functions. Generate the first few series expansion coefficients of a function:

Define a utility function for building Hankel determinants out of series coefficients:

Generate the first few coefficients for a continued fraction expansion:

Construct the continued fraction approximation:

Compare the continued fraction with the original function:

The n-point Gaussian quadrature rule corresponding to a weight function over the interval can be derived from the moments of the weight function. Define a weight function and an interval:

Generate the first moments starting from 0:

The coefficients of the n-order orthogonal polynomial corresponding to the given weight function can be obtained by solving a Hankel system constructed from the moments:

The nodes of the n-point Gaussian quadrature rule are the roots of the orthogonal polynomial:

The weights of the n-point Gaussian quadrature rule can be obtained by solving a Vandermonde system:

Use the nodes and weights to numerically approximate an integral:

Compare with the answer obtained from NIntegrate:

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:

Define a Hankel matrix whose entries are the reciprocals of the factorials:

Its determinant can be expressed in terms of the Barnes G-function:

Define a Hankel matrix whose entries are the Bernoulli numbers:

Its determinant can be expressed in terms of the Barnes G-function: