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
We consider the following sequence of 2n-1 numbers
| > |
seq(a[n],n=0..3),__,a[2*n-2]; |
![a[0], a[1], a[2], a[3], __, a[`+`(`*`(2, `*`(n)), `-`(2))]](/view.aspx?SI=5607/HankelMatrix_2.gif) |
(1) |
which may be the coefficients of a quadratic form in n variables:
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Q(x,x):=sum(sum(a[i+k]*x[i]*x[k],i=0..n-1),k=0..n-1); |
![`:=`(Q(x, x), sum(sum(`*`(a[`+`(i, k)], `*`(x[i], `*`(x[k]))), i = 0 .. `+`(n, `-`(1))), k = 0 .. `+`(n, `-`(1))))](/view.aspx?SI=5607/HankelMatrix_3.gif) |
(2) |
This is called a Hankel form. The symmetric matrix
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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]; |
![D[1], D[2], D[3], __, D[N]](/view.aspx?SI=5607/HankelMatrix_6.gif) |
(5) |
If the first j rows of the Hankel matrix are linear independent, but the first j+1 rows
linear dependent, then
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D[j]=not_equal_to_zero; |
![D[j] = not_equal_to_zero](/view.aspx?SI=5607/HankelMatrix_7.gif) |
(6) |
Examples:
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Hankel[3*columns]:=HankelMatrix([seq(a[n],n=0..4)]); |
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(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) |
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(8) |
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(8) |
| > |
Hankel[4*rows]:=HankelMatrix([seq(a[n],n=0..6)]); |
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(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; |
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(10) |
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(10) |
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(10) |
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(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]; |
![R[1], R[2], R[3], __, R[`+`(j, 1)]](/view.aspx?SI=5607/HankelMatrix_17.gif) |
(11) |
where
| > |
seq(R[q],q=1..3),__,R[j]; |
![R[1], R[2], R[3], __, R[j]](/view.aspx?SI=5607/HankelMatrix_18.gif) |
(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); |
![`:=`(R[`+`(j, 1)], sum(`*`(c[p], `*`(R[`+`(j, `-`(p), 1)])), p = 1 .. j))](/view.aspx?SI=5607/HankelMatrix_19.gif) |
(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:
| > |
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.
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for p in [1,3,5,7,9] do
Hankel_matrix[p*different_elements]:= HankelMatrix([seq(a[n],n=0..p-1)])
od; |
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(16) |
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(16) |
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(16) |
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(16) |
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(16) |
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; |
![`:=`(rank[different_elements], 1)](/view.aspx?SI=5607/HankelMatrix_27.gif) |
(17) |
![`:=`(rank[`+`(`*`(3, `*`(different_elements)))], 2)](/view.aspx?SI=5607/HankelMatrix_28.gif) |
(17) |
![`:=`(rank[`+`(`*`(5, `*`(different_elements)))], 3)](/view.aspx?SI=5607/HankelMatrix_29.gif) |
(17) |
![`:=`(rank[`+`(`*`(7, `*`(different_elements)))], 4)](/view.aspx?SI=5607/HankelMatrix_30.gif) |
(17) |
![`:=`(rank[`+`(`*`(9, `*`(different_elements)))], 5)](/view.aspx?SI=5607/HankelMatrix_31.gif) |
(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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