Editing Hankel matrix (section)

==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<ref>{{harvnb|Fuhrmann|2012|loc=§8.3}}</ref>
<math display="block">
 H_f : \mathbf C[z] \to \mathbf z^{-1} \mathbf C[[z^{-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 C[[z^{-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 C[[z^{-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 (linear algebra)|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>