Power of two

In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a square matrix with constant skew-diagonals (positive sloping diagonals), e.g.:

$\left[\begin{array}{ccccc}a& b& c& d& e\\ b& c& d& e& f\\ c& d& e& f& g\\ d& e& f& g& h\\ e& f& g& h& i\\ \end{array}\right].$

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

$A_{i,j}=A_{i-1,j+1}.$

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

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

$c_n=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})b_k$

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

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 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

Positive Hankel matrices and the Hamburger moment problems

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See also

• Hamburger moment problem
• Toeplitz matrix, a Hankel matrix 'upside-down'.

References

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