Hankel matrix

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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.:

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 \\

Any n×n matrix A of the form

A =
  a_{0} & a_{1} & a_{2} & \ldots & \ldots  &a_{n-1}  \\
  a_{1} & a_2 &  &  & &\vdots \\
  a_{2} &  &  & & & \vdots \\ 
 \vdots & & & &  & a_{2n-4}\\
 \vdots & &  & & a_{2n-4}&  a_{2n-3} \\
a_{n-1} &  \ldots & \ldots & a_{2n-4} & a_{2n-3} & a_{2n-2}

is a Hankel matrix. If the i,j element of A is denoted Ai,j, then we have

A_{i,j} = A_{i+1,j-1} = a_{i+j-2}.\

The Hankel matrix is closely related to the Toeplitz matrix (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 is one whose matrix with respect to an orthonormal basis is a (possibly infinite) Hankel matrix (A_{i,j})_{i,j \ge 1}, where  A_{i,j} depends only on i+j.

The determinant of a Hankel matrix is called a catalecticant.

Hankel transform[edit]

The Hankel transform is the name sometimes given to the transformation of a sequence, where the transformed sequence corresponds to the determinant of the Hankel matrix. That is, the sequence \{h_n\}_{n\ge 0} is the Hankel transform of the sequence \{b_n\}_{n\ge 0} when

h_n = \det (b_{i+j-2})_{1 \le i,j \le n+1}.

Here, a_{i,j}=b_{i+j-2} is the Hankel matrix of the sequence \{b_n\}. The Hankel transform is invariant under the binomial transform of a sequence. That is, if one writes

c_n = \sum_{k=0}^n {n \choose k} b_k

as the binomial transform of the sequence \{b_n\}, then one has

\det (b_{i+j-2})_{1 \le i,j \le n+1} = \det (c_{i+j-2})_{1 \le i,j \le n+1}.

Hankel matrices for system identification[edit]

Hankel matrices are formed when, given a sequence of output data, a realization of an underlying state-space or hidden Markov model is desired. The singular value decomposition of the Hankel matrix provides a means of computing the A, B, and C matrices which define the state-space realization.

Orthogonal polynomials on the real line[edit]

Positive Hankel matrices and the Hamburger moment problems[edit]

Further information: Hamburger moment problem

Orthogonal polynomials on the real line[edit]

Tridiagonal model of positive Hankel operators[edit]

Relation between Hankel and Toeplitz matrices[edit]

Let  J_n be the reflection matrix of order  n . For example the reflection matrix of order  5 is as follows:  J_5 = \begin{bmatrix}
 &  &  &  & 1 \\
 &  &  & 1 &  \\
 &  & 1 &  &  \\
 & 1 &  &  &  \\
1 &  &  &  &  \\

If  H(m,n) is a m \times n Hankel matrix, then  H(m,n) = T(m, n) \, J_n , where  T(m,n) is a m \times n Toeplitz matrix.

Relations between structured matrices[edit]

See also[edit]