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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.:

Contents

[ 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 … … a n − 1 a 1 a 2 ⋮ a 2 ⋮ ⋮ a 2 n − 4 ⋮ a 2 n − 4 a 2 n − 3 a n − 1 … … a 2 n − 4 a 2 n − 3 a 2 n − 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 ≥ 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 ≥ 0 is the Hankel transform of the sequence { b n } n ≥ 0 when

h n = det ( b i + j − 2 ) 1 ≤ i , j ≤ 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 = ∑ k = 0 n ( n k ) b k

as the binomial transform of the sequence { b n } , then one has

det ( b i + j − 2 ) 1 ≤ i , j ≤ n + 1 = det ( c i + j − 2 ) 1 ≤ i , j ≤ n + 1 .

Applications of Hankel matrices

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. The Hankel matrix formed from the signal has been found useful for decomposition of non-stationary signals.

Relation between Hankel and Toeplitz matrices

Let J n be the reflection matrix of order n . For example the reflection matrix of order 5 is as follows: J 5 = [ 1 1 1 1 1 ] .

If H ( m , n ) is a m × n Hankel matrix, then H ( m , n ) = T ( m , n ) J n , where T ( m , n ) is a m × n Toeplitz matrix.

References

Hankel matrix Wikipedia
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