= Hankel matrix
{c}
{title2=$H_{ij}=a_{i+j}$}
A <Hankel matrix> has entries constant on each anti-diagonal: with indices starting at zero, $H_{ij}=a_{i+j}$. For example,
$$
H=\begin{pmatrix}a_0&a_1&a_2\\a_1&a_2&a_3\\a_2&a_3&a_4\end{pmatrix}.
$$
This differs from a <Toeplitz matrix>, whose entries are constant along diagonals. A square <Hankel matrix> is symmetric, although it need not be positive semidefinite.
An important example uses moments $a_k=\int x^k\,d\mu(x)$ of a positive <measure> on the <real line>, assuming the needed moments are finite. For a real coefficient <vector> $c$,
$$
c^THc=\int\left(\sum_i c_ix^i\right)^2d\mu(x)\ge0,
$$
so the moment <Hankel matrix> is <positive semidefinite>. Taking the point mass at $x=2$ gives $a_k=2^k$ and $H=(1,2,4)^T(1,2,4)$, of rank one. Thus the matrix pattern connects sequences, <polynomials> and <moment problems>.
Back to article page
