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,

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

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

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

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

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

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 ${\displaystyle A}$ 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.