Hankel matrix

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,

$${\displaystyle \qquad {\begin{bmatrix}a&b&c&d&e\\b&c&d&e&f\\c&d&e&f&g\\d&e&f&g&h\\e&f&g&h&i\\\end{bmatrix}}.}$${\displaystyle \qquad {\begin{bmatrix}a&b&c&d&e\\b&c&d&e&f\\c&d&e&f&g\\d&e&f&g&h\\e&f&g&h&i\\\end{bmatrix}}.}

More generally, a Hankel matrix is any ${\displaystyle n\times n}${\displaystyle n\times n} matrix ${\displaystyle A}${\displaystyle A} of the form

$${\displaystyle A={\begin{bmatrix}a_{0}&a_{1}&a_{2}&\ldots &a_{n-1}\\a_{1}&a_{2}&&&\vdots \\a_{2}&&&&a_{2n-4}\\\vdots &&&a_{2n-4}&a_{2n-3}\\a_{n-1}&\ldots &a_{2n-4}&a_{2n-3}&a_{2n-2}\end{bmatrix}}.}$${\displaystyle A={\begin{bmatrix}a_{0}&a_{1}&a_{2}&\ldots &a_{n-1}\\a_{1}&a_{2}&&&\vdots \\a_{2}&&&&a_{2n-4}\\\vdots &&&a_{2n-4}&a_{2n-3}\\a_{n-1}&\ldots &a_{2n-4}&a_{2n-3}&a_{2n-2}\end{bmatrix}}.}

In terms of the components, if the ${\displaystyle i,j}${\displaystyle i,j} element of ${\displaystyle A}${\displaystyle A} is denoted with ${\displaystyle A_{ij}}${\displaystyle A_{ij}}, and assuming ${\displaystyle i\leq j}${\displaystyle i\leq j}, then we have ${\displaystyle A_{i,j}=A_{i+k,j-k}}${\displaystyle A_{i,j}=A_{i+k,j-k}} for all ${\displaystyle k=0,...,j-i.}${\displaystyle k=0,...,j-i.}

• Any square Hankel matrix is symmetric.
• Let ${\displaystyle J_{n}}${\displaystyle J_{n}} be the ${\displaystyle n\times n}${\displaystyle n\times n} exchange matrix. If ${\displaystyle H}${\displaystyle H} is an ${\displaystyle m\times n}${\displaystyle m\times n} Hankel matrix, then ${\displaystyle H=TJ_{n}}${\displaystyle H=TJ_{n}} where ${\displaystyle T}${\displaystyle T} is an ${\displaystyle m\times n}${\displaystyle m\times n} Toeplitz matrix.
  • If ${\displaystyle T}${\displaystyle T} is real symmetric, then ${\displaystyle H=TJ_{n}}${\displaystyle H=TJ_{n}} will have the same eigenvalues as ${\displaystyle T}${\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.

Given a formal Laurent series $${\displaystyle f(z)=\sum _{n=-\infty }^{N}a_{n}z^{n},}$${\displaystyle f(z)=\sum _{n=-\infty }^{N}a_{n}z^{n},} the corresponding Hankel operator is defined as^[2] $${\displaystyle H_{f}:\mathbf {C} [z]\to \mathbf {z} ^{-1}\mathbf {C} [[z^{-1}]].}$${\displaystyle H_{f}:\mathbf {C} [z]\to \mathbf {z} ^{-1}\mathbf {C} [[z^{-1}]].} This takes a polynomial ${\displaystyle g\in \mathbf {C} [z]}${\displaystyle g\in \mathbf {C} [z]} and sends it to the product ${\displaystyle fg}${\displaystyle fg}, but discards all powers of ${\displaystyle z}${\displaystyle z} with a non-negative exponent, so as to give an element in ${\displaystyle z^{-1}\mathbf {C} [[z^{-1}]]}${\displaystyle z^{-1}\mathbf {C} [[z^{-1}]]}, the formal power series with strictly negative exponents. The map ${\displaystyle H_{f}}${\displaystyle H_{f}} is in a natural way ${\displaystyle \mathbf {C} [z]}${\displaystyle \mathbf {C} [z]}-linear, and its matrix with respect to the elements ${\displaystyle 1,z,z^{2},\dots \in \mathbf {C} [z]}${\displaystyle 1,z,z^{2},\dots \in \mathbf {C} [z]} and ${\displaystyle z^{-1},z^{-2},\dots \in z^{-1}\mathbf {C} [[z^{-1}]]}${\displaystyle z^{-1},z^{-2},\dots \in z^{-1}\mathbf {C} [[z^{-1}]]} is the Hankel matrix $${\displaystyle {\begin{bmatrix}a_{1}&a_{2}&\ldots \\a_{2}&a_{3}&\ldots \\a_{3}&a_{4}&\ldots \\\vdots &\vdots &\ddots \end{bmatrix}}.}$${\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 ${\displaystyle f}${\displaystyle f} is a rational function, that is, a fraction of two polynomials $${\displaystyle f(z)={\frac {p(z)}{q(z)}}.}$${\displaystyle f(z)={\frac {p(z)}{q(z)}}.}

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 ${\displaystyle A}${\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.

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

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.^[5]

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

• 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.

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