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,
<math display=block>\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}.</math>
More generally, a Hankel matrix is any <math>n \times n</math> matrix <math>A</math> of the form
<math display=block>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}.</math>
In terms of the components, if the <math>i,j</math> element of <math>A</math> is denoted with <math>A_{ij}</math>, and assuming <math>i \le j</math>, then we have <math>A_{i,j} = A_{i+k,j-k}</math> for all <math>k = 0,...,j-i.</math>
Properties
- Any square Hankel matrix is symmetric.
- Let <math>J_n</math> be the <math>n \times n</math> exchange matrix. If <math>H</math> is an <math>m \times n</math> Hankel matrix, then <math>H = T J_n</math> where <math>T</math> is an <math>m \times n</math> Toeplitz matrix.
- If <math>T</math> is real symmetric, then <math>H = T J_n</math> will have the same eigenvalues as <math>T</math> up to sign.
- The Hilbert matrix is an example of a Hankel matrix.
- The determinant of a Hankel matrix is called a catalecticant.
- If <math>H</math> is an <math>n \times n</math> Hankel matrix, then <math>H=V^TDV</math> where <math>V</math> is a confluent Vandermonde matrix and <math>D</math> is a block diagonal matrix, with symmetric and upper anti-triangular blocks.
Hankel operator
Given a formal Laurent series
<math display="block">
f(z) = \sum_{n=-\infty}^N a_n z^n,
</math>
the corresponding Hankel operator is defined as
<math display="block">
H_f : \mathbf C[z] \to \mathbf z^{-1} \mathbf Cz^{-1}.
</math>
This takes a polynomial <math>g \in \mathbf C[z]</math> and sends it to the product <math>fg</math>, but discards all powers of <math>z</math> with a non-negative exponent, so as to give an element in <math>z^{-1} \mathbf Cz^{-1}</math>, the formal power series with strictly negative exponents. The map <math>H_f</math> is in a natural way <math>\mathbf C[z]</math>-linear, and its matrix with respect to the elements <math>1, z, z^2, \dots \in \mathbf C[z]</math> and <math>z^{-1}, z^{-2}, \dots \in z^{-1}\mathbf Cz^{-1}</math> is the Hankel matrix
<math display=block>\begin{bmatrix}
a_{-1} & a_{-2} & \ldots \\
a_{-2} & a_{-3} & \ldots \\
a_{-3} & a_{-4} & \ldots \\
\vdots & \vdots & \ddots
\end{bmatrix}.</math>
Any Hankel matrix arises in this way. A theorem due to Kronecker says that the rank of this matrix is finite precisely if <math>f</math> is a rational function, that is, a fraction of two polynomials
<math display="block">
f(z) = \frac{p(z)}{q(z)}.
</math>
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 <math>A</math> 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 <math>b_k</math> is the sequence of the determinants of the Hankel matrices formed from <math>b_k</math>. Given an integer <math>n > 0</math>, define the corresponding <math>(n \times n)</math>-dimensional Hankel matrix <math>B_n</math> as having the matrix elements <math>[B_n]_{i,j} = b_{i+j}.</math> Then the sequence <math>h_n</math> given by
<math display="block">
h_n = \det B_n
</math>
is the Hankel transform of the sequence <math>b_k.</math> The Hankel transform is invariant under the binomial transform of a sequence. That is, if one writes
<math display="block">
c_n = \sum_{k=0}^n {n \choose k} b_k
</math>
as the binomial transform of the sequence <math>b_n</math>, then one has <math>\det B_n = \det C_n.</math>
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 (editors—T. Kailath, A.H. Sayed), ch.4 (SIAM).