In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a rectangular matrix in which each ascending skew-diagonal from left to right is constant. For example,

More generally, a Hankel matrix matrix of the form

Properties

• Any square Hankel matrix is symmetric.

• exchange matrix Toeplitz matrix.

• is real will have the same eigenvalues up to sign.

• The Hilbert matrix is an example of a Hankel matrix.

• The determinant of a Hankel matrix is called a catalecticant.

• is a confluent Vandermonde matrix is a block diagonal matrix, with symmetric and upper anti-triangular blocks.

Hankel operator

Given a formal Laurent series the corresponding Hankel operator is defined as This takes a polynomial, the formal power series Any Hankel matrix arises in this way. A theorem due to Kronecker says that the rank is a rational function

Approximations

We are often interested in approximations of the Hankel operators, possibly by low-order operators. In order to approximate the output of the operator, we can use the spectral norm (operator 2-norm) to measure the error of our approximation. This suggests singular value decomposition as a possible technique to approximate the action of the operator.

does not have to be finite. If it is infinite, traditional methods of computing individual singular vectors will not work directly. We also require that the approximation is a Hankel matrix, which can be shown with AAK theory.

Hankel matrix transform

The Hankel matrix transform, or simply Hankel transform, of a sequence The Hankel transform is invariant under the binomial transform

Applications of Hankel matrices

Hankel matrices are formed when, given a sequence of output data, a realization of an underlying state-space or hidden Markov model is desired. The singular value decomposition of the Hankel matrix provides a means of computing the A, B, and C matrices which define the state-space realization. The Hankel matrix formed from the signal has been found useful for decomposition of non-stationary signals and time-frequency representation.

Method of moments for polynomial distributions

The method of moments applied to polynomial distributions results in a Hankel matrix that needs to be inverted in order to obtain the weight parameters of the polynomial distribution approximation.

Positive Hankel matrices and the Hamburger moment problems

See also

• Cauchy matrix

• Jacobi operator

• Toeplitz matrix, an "upside down" (that is, row-reversed) Hankel matrix

• Vandermonde matrix

Notes

References

• Brent R.P. (1999), "Stability of fast algorithms for structured linear systems", Fast Reliable Algorithms for Matrices with Structure (editorsT. Kailath, A.H. Sayed), ch.4 (SIAM).