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.
More generally, a Hankel matrix is any
In terms of the components, if the
Given a formal Laurent series
the corresponding Hankel operator is defined as[2]
This takes a polynomial
and sends it to the product
, but discards all powers of
with a non-negative exponent, so as to give an element in
, the formal power series with strictly negative exponents.
-linear, and its matrix with respect to the elements
is the Hankel matrix
Any Hankel matrix arises in this way.
A theorem due to Kronecker says that the rank of this matrix is finite precisely if
is a rational function, that is, a fraction of two polynomials
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
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 Hankel matrix transform, or simply Hankel transform, of a sequence
is the sequence of the determinants of the Hankel matrices formed from
is the Hankel transform of the sequence
The Hankel transform is invariant under the binomial transform of a sequence.
as the binomial transform of the sequence
Hankel matrices are formed when, given a sequence of output data, a realization of an underlying state-space or hidden Markov model is desired.
[3] 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.
[4] The Hankel matrix formed from the signal has been found useful for decomposition of non-stationary signals and time-frequency representation.
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.