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.:

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

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 $(A_{i,j})_{i,j\geq 1}$, 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 $\{h_{n}\}_{n\geq 0}$ is the Hankel transform of the sequence $\{b_{n}\}_{n\geq 0}$ when

$h_{n}=\det(b_{i+j-2})_{1\leq i,j\leq n+1}.$

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

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

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

47 year-old Podiatrist Hyslop from Alert Bay, has lots of hobbies and interests that include fencing, property developers in condo new launch singapore and handball. Just had a family trip to Monasteries of Haghpat and Sanahin.

See also

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

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

Template:Linear-algebra-stub