Hankel matrix

In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a square matrix in which each ascending skew-diagonal from left to right is constant, e.g.:
More generally, a Hankel matrix is any formula_2 matrix formula_3 of the form
In terms of the components, if the formula_5 element of formula_3 is denoted with formula_7, and assuming formula_8, then we have formula_9 for all formula_10
Hankel operator.
A Hankel operator on a Hilbert space is one whose matrix is a (possibly infinite) Hankel matrix with respect to an orthonormal basis. As indicated above, a Hankel Matrix is a matrix with constant values along its antidiagonals, which means that a Hankel matrix formula_21 must satisfy, for all rows formula_22 and columns formula_23, formula_24. Note that every entry formula_25 depends only on formula_26.
Let the corresponding Hankel Operator be formula_27. Given a Hankel matrix formula_3, the corresponding Hankel operator is then defined as formula_29. 
We are often interested in Hankel operators formula_30 over the Hilbert space formula_31, the space of square integrable bilateral complex sequences. For any formula_32, we have
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.
Note that the matrix formula_3 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.
The determinant of a Hankel matrix is called a catalecticant.
Hankel matrix transform.
The Hankel matrix transform, or simply Hankel transform, produces the sequence of the determinants of the Hankel matrices formed from the given sequence. Namely, the sequence formula_35 is the Hankel transform of the sequence formula_36 when
The Hankel transform is invariant under the binomial transform of a sequence. That is, if one writes
as the binomial transform of the sequence formula_39, then one has
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.