In linear algebra

Linear algebra

Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel

Hermann Hankel

Hermann Hankel was a German mathematician who was born in Halle, Germany and died in Schramberg , Imperial Germany....

, is a square matrix with constant skew-diagonals (positive sloping diagonals), e.g.:


If the i,j element of A is denoted A_i,j, then we have


The Hankel matrix is closely related to the Toeplitz matrix

Toeplitz matrix

In linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant...

 (a Hankel matrix is an upside-down Toeplitz matrix). For a special case of this matrix see Hilbert matrix.

A Hankel operator on a Hilbert space

Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

 is one whose matrix with respect to an orthonormal basis

Orthonormal basis

In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...

 is an infinite Hankel matrix
, where depends only on .

The determinant of a Hankel matrix is called a catalecticant

Catalecticant

In mathematical invariant theory, the catalecticant of a binary form of degree 2n is a polynomial in its coefficients that vanishes when the binary form is a sum of at most n powers . It was introduced by ; see...

.

Hankel transform

The Hankel transform is the name sometimes given to the transformation of a sequence

Sequence

In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

, where the transformed sequence corresponds to the determinant of the Hankel matrix. That is, the sequence is the Hankel transform of the sequence when


Here, is the Hankel matrix of the sequence . The Hankel transform is invariant under the binomial transform of a sequence. That is, if one writes


as the binomial transform of the sequence , then one has

Hankel matrices for system identification

System identification

In control engineering, the field of system identification uses statistical methods to build mathematical models of dynamical systems from measured data...

 

Hankel matrices are formed when given a sequence of output data and a realization of an underlying state-space or hidden Markov model is desired. The singular value decomposition

Singular value decomposition

In linear algebra, the singular value decomposition is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics....

of the Hankel matrix provides a means of computing the A, B, and C matrices which define the state-space realization.