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# Hankel Matrix

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Hankel Matrix

Univ.-Prof. Dr.-Ing. habil. Josef  BETTEN
RWTH University Aachen
Mathematical Models in Materials Science and Continuum Mechanics
Augustinerbach 4-20
D-52056  A a c h e n ,  Germany
betten@mmw.rwth-aachen.de

Abstract

Using MAPLE 11, properties of  the Hankel Matrix have been discussed and some representations have been proposed.

Keywords:  Hankel form and matrix; principal minors; forming of  Hankel matrices

Definitions

 > restart:

 > with(LinearAlgebra):

We consider the following sequence of  2n-1 numbers

 > seq(a[n],n=0..3),__,a[2*n-2];

 (1)

which may be the coefficients of  a quadratic form in  n  variables:

 > Q(x,x):=sum(sum(a[i+k]*x[i]*x[k],i=0..n-1),k=0..n-1);

 (2)

This is called a Hankel form. The symmetric matrix

 > Matrix([a[i+k],i=0..n-1,k=0..n-1]);

 (3)

corresponding to that form is called a Hankel matrix. It can be written as:

 > A[Hankel]:=Matrix([[seq(a[n],n=0..3),__,a[N-1]], [seq(a[n],n=1..4),__,a[N]], [seq(a[n],n=2..5),__,a[N+1]], [seq(a[n],n=3..6),__,a[N+2]], [???,???,???,???,???,???], [a[N-1],a[N],a[N+1],a[N+2],__,a[2*N-2]]]);

 (4)

Principal Minors

The sequence of  principal minors of  the Hankel matrix is denoted by

 > seq(D[q],q=1..3),__,D[N];

 (5)

If the first  j  rows of  the Hankel matrix are linear independent, but the first  j+1  rows

linear dependent, then

 > D[j]=not_equal_to_zero;

 (6)

Examples:

 > Hankel[3*columns]:=HankelMatrix([seq(a[n],n=0..4)]);

 (7)

 > for i in [1,2,3] do principal_minor[i,i]:=Minor(Hankel[3*columns],i,i, output=['matrix','determinant'],method='minor') od;

 (8)

 (8)

 (8)

 > Hankel[4*rows]:=HankelMatrix([seq(a[n],n=0..6)]);

 (9)

 > for i in [1,2,3,4] do principal_minor[i,i]:=Minor(Hankel[4*rows],i,i, output=['matrix','determinant'],method='minor') od;

 (10)

 (10)

 (10)

 (10)

The Hankel Matrix formed from the first j Rows

The first j+1 rows of the Hankel matrix are denoted by

 > seq(R[q],q=1..3),__,R[j+1];

 (11)

where

 > seq(R[q],q=1..3),__,R[j];

 (12)

are linearly indepedent and R[j+1] is expressed linearly in terms of  them:

 > R[j+1]:=sum(c[p]*R[j-p+1],p=1..j);

 (13)

The matrix formed from the first j rows is:

 > M[R[1]..R[j]]:=Matrix([[a[0],a[1],a[2],__,a[N-1]], [a[1],a[2],a[3],__,a[N]], [???,???,???,???,???], [a[j-1],a[j],a[j+1],__,a[j+N-2]]]);

 (14)

This matrix is of rank  J < N.

Construction of  Hankel Matrices

Hankel matrices are symmetric and can be constructed, for instance, in the

following ways:

 > restart:

 > with(LinearAlgebra):

 > HANKEL[N=1..5]:=seq(Matrix(N,N,(i,k)->a[i-1+k-1]),N=1..5);

 (15)

where  N  is the number of columns or row. There are an odd number

of   p = 2*N-1  different elements.

 > restart:

 > with(LinearAlgebra):

 > for p in [1,3,5,7,9] do Hankel_matrix[p*different_elements]:= HankelMatrix([seq(a[n],n=0..p-1)]) od;

 (16)

 (16)

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These matrices are of rank N, if  p = 2*N-1  is the odd number of different elements:

 > for p in [1,3,5,7,9] do rank[p*different_elements]:= Rank(HankelMatrix([seq(a[n],n=0..p-1)])) od;

 (17)

 (17)

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 (17)

Further properties of  the Hankel Matrix and other forms have been discussed in:

Gantmacher, F.R.: The Theory of Matrices, Volume I and II, Chelsea Publishing Company,  New York 1977.

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