• Images
• Videos
• Documents
• Books
• Articles

Properties

• Any Hankel matrix is symmetric.
• Let J n {\displaystyle J_{n}} be the n × n {\displaystyle n\times n} exchange matrix. If H {\displaystyle H} is an m × n {\displaystyle m\times n} Hankel matrix, then H = T J n {\displaystyle H=TJ_{n}} where T {\displaystyle T} is an m × n {\displaystyle m\times n} Toeplitz matrix.
  • If T {\displaystyle T} is real symmetric, then H = T J n {\displaystyle H=TJ_{n}} will have the same eigenvalues as T {\displaystyle T} up to sign.¹
• The Hilbert matrix is an example of a Hankel matrix.
• The determinant of a Hankel matrix is called a catalecticant.

Hankel operator

Given a formal Laurent series f ( z ) = ∑ n = − ∞ N a n z n , {\displaystyle f(z)=\sum _{n=-\infty }^{N}a_{n}z^{n},} the corresponding Hankel operator is defined as² H f : C [ z ] → z − 1 C [ [ z − 1 ] ] . {\displaystyle H_{f}:\mathbf {C} [z]\to \mathbf {z} ^{-1}\mathbf {C} [[z^{-1}]].} This takes a polynomial g ∈ C [ z ] {\displaystyle g\in \mathbf {C} [z]} and sends it to the product f g {\displaystyle fg} , but discards all powers of z {\displaystyle z} with a non-negative exponent, so as to give an element in z − 1 C [ [ z − 1 ] ] {\displaystyle z^{-1}\mathbf {C} [[z^{-1}]]} , the formal power series with strictly negative exponents. The map H f {\displaystyle H_{f}} is in a natural way C [ z ] {\displaystyle \mathbf {C} [z]} -linear, and its matrix with respect to the elements 1 , z , z 2 , ⋯ ∈ C [ z ] {\displaystyle 1,z,z^{2},\dots \in \mathbf {C} [z]} and z − 1 , z − 2 , ⋯ ∈ z − 1 C [ [ z − 1 ] ] {\displaystyle z^{-1},z^{-2},\dots \in z^{-1}\mathbf {C} [[z^{-1}]]} is the Hankel matrix [ a 1 a 2 … a 2 a 3 … a 3 a 4 … ⋮ ⋮ ⋱ ] . {\displaystyle {\begin{bmatrix}a_{1}&a_{2}&\ldots \\a_{2}&a_{3}&\ldots \\a_{3}&a_{4}&\ldots \\\vdots &\vdots &\ddots \end{bmatrix}}.} Any Hankel matrix arises in this way. A theorem due to Kronecker says that the rank of this matrix is finite precisely if f {\displaystyle f} is a rational function, that is, a fraction of two polynomials f ( z ) = p ( z ) q ( z ) . {\displaystyle f(z)={\frac {p(z)}{q(z)}}.}

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.

Note that the matrix A {\displaystyle A} 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

Not to be confused with Hankel transform.

The Hankel matrix transform, or simply Hankel transform, of a sequence b k {\displaystyle b_{k}} is the sequence of the determinants of the Hankel matrices formed from b k {\displaystyle b_{k}} . Given an integer n > 0 {\displaystyle n>0} , define the corresponding ( n × n ) {\displaystyle (n\times n)} -dimensional Hankel matrix B n {\displaystyle B_{n}} as having the matrix elements [ B n ] i , j = b i + j . {\displaystyle [B_{n}]_{i,j}=b_{i+j}.} Then the sequence h n {\displaystyle h_{n}} given by h n = det B n {\displaystyle h_{n}=\det B_{n}} is the Hankel transform of the sequence b k . {\displaystyle b_{k}.} The Hankel transform is invariant under the binomial transform of a sequence. That is, if one writes c n = ∑ k = 0 n ( n k ) b k {\displaystyle c_{n}=\sum _{k=0}^{n}{n \choose k}b_{k}} as the binomial transform of the sequence b n {\displaystyle b_{n}} , then one has det B n = det C n . {\displaystyle \det B_{n}=\det C_{n}.}

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.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.

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.⁵

Positive Hankel matrices and the Hamburger moment problems

Further information: Hamburger moment problem

See also

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

Notes

• Brent R.P. (1999), "Stability of fast algorithms for structured linear systems", Fast Reliable Algorithms for Matrices with Structure (editors—T. Kailath, A.H. Sayed), ch.4 (SIAM).
• Fuhrmann, Paul A. (2012). A polynomial approach to linear algebra. Universitext (2 ed.). New York, NY: Springer. doi:10.1007/978-1-4614-0338-8. ISBN 978-1-4614-0337-1. Zbl 1239.15001.

• Victor Y. Pan (2001). Structured matrices and polynomials: unified superfast algorithms. Birkhäuser. ISBN 0817642404.
• J.R. Partington (1988). An introduction to Hankel operators. LMS Student Texts. Vol. 13. Cambridge University Press. ISBN 0-521-36791-3.

References

1. Yasuda, M. (2003). "A Spectral Characterization of Hermitian Centrosymmetric and Hermitian Skew-Centrosymmetric K-Matrices". SIAM J. Matrix Anal. Appl. 25 (3): 601–605. doi:10.1137/S0895479802418835. /wiki/Doi_(identifier) ↩

2. Fuhrmann 2012, §8.3 - Fuhrmann, Paul A. (2012). A polynomial approach to linear algebra. Universitext (2 ed.). New York, NY: Springer. doi:10.1007/978-1-4614-0338-8. ISBN 978-1-4614-0337-1. Zbl 1239.15001. https://doi.org/10.1007%2F978-1-4614-0338-8 ↩

3. Aoki, Masanao (1983). "Prediction of Time Series". Notes on Economic Time Series Analysis : System Theoretic Perspectives. New York: Springer. pp. 38–47. ISBN 0-387-12696-1. 0-387-12696-1 ↩

4. Aoki, Masanao (1983). "Rank determination of Hankel matrices". Notes on Economic Time Series Analysis : System Theoretic Perspectives. New York: Springer. pp. 67–68. ISBN 0-387-12696-1. 0-387-12696-1 ↩

5. J. Munkhammar, L. Mattsson, J. Rydén (2017) "Polynomial probability distribution estimation using the method of moments". PLoS ONE 12(4): e0174573. https://doi.org/10.1371/journal.pone.0174573 https://doi.org/10.1371/journal.pone.0174573 ↩