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Concept
Hankel matrix
Summary
In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a square matrix in which each ascending skew-diagonal from left to right is constant, e.g.:
More generally, a Hankel matrix is any matrix of the form
In terms of the components, if the element of is denoted with , and assuming , then we have for all
Any Hankel matrix is symmetric.
Let be the exchange matrix. If is a Hankel matrix, then where is a Toeplitz matrix.
If is real symmetric, then will have the same eigenvalues as up to sign.
The Hilbert matrix is an example of a Hankel matrix.
Hankel matrices are closely related to formal Laurent series. In fact, such a series gives rise to a linear map, referred to as a Hankel operator
which takes a polynomial and sends it to the product , but discards all powers of with a non-negative exponent, so as to give an element in , the formal power series with strictly negative exponents. The map is in a natural way -linear, and its matrix with respect to the elements and is the Hankel matrix
Any Hankel matrix arises in such a way. A theorem due to Kronecker says that the rank of this matrix is finite precisely if is a rational function, i.e., a fraction of two polynomials .
A Hankel operator on a Hilbert space is one whose matrix is a (possibly infinite) Hankel matrix with respect to an orthonormal basis. As indicated above, a Hankel Matrix is a matrix with constant values along its antidiagonals, which means that a Hankel matrix must satisfy, for all rows and columns , . Note that every entry depends only on .
Let the corresponding Hankel Operator be . Given a Hankel matrix , the corresponding Hankel operator is then defined as .
We are often interested in Hankel operators over the Hilbert space , the space of square integrable bilateral complex sequences. For any , we have
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.
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