- Source: Hankel matrix
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. For example,
[
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
]
.
{\displaystyle \qquad {\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}}.}
More generally, a Hankel matrix is any
n
×
n
{\displaystyle n\times n}
matrix
A
{\displaystyle 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
]
.
{\displaystyle A={\begin{bmatrix}a_{0}&a_{1}&a_{2}&\ldots &a_{n-1}\\a_{1}&a_{2}&&&\vdots \\a_{2}&&&&a_{2n-4}\\\vdots &&&a_{2n-4}&a_{2n-3}\\a_{n-1}&\ldots &a_{2n-4}&a_{2n-3}&a_{2n-2}\end{bmatrix}}.}
In terms of the components, if the
i
,
j
{\displaystyle i,j}
element of
A
{\displaystyle A}
is denoted with
A
i
j
{\displaystyle A_{ij}}
, and assuming
i
≤
j
{\displaystyle i\leq j}
, then we have
A
i
,
j
=
A
i
+
k
,
j
−
k
{\displaystyle A_{i,j}=A_{i+k,j-k}}
for all
k
=
0
,
.
.
.
,
j
−
i
.
{\displaystyle k=0,...,j-i.}
Properties
Any Hankel matrix is symmetric.
Let
J
n
{\displaystyle J_{n}}
be the
n
×
n
{\displaystyle n\times n}
exchange matrix. If
H
{\displaystyle H}
is an
m
×
n
{\displaystyle m\times n}
Hankel matrix, then
H
=
T
J
n
{\displaystyle H=TJ_{n}}
where
T
{\displaystyle T}
is an
m
×
n
{\displaystyle m\times n}
Toeplitz matrix.
If
T
{\displaystyle T}
is real symmetric, then
H
=
T
J
n
{\displaystyle H=TJ_{n}}
will have the same eigenvalues as
T
{\displaystyle T}
up to sign.
The Hilbert matrix is an example of a Hankel matrix.
The determinant of a Hankel matrix is called a catalecticant.
Hankel operator
Given a formal Laurent series
f
(
z
)
=
∑
n
=
−
∞
N
a
n
z
n
,
{\displaystyle f(z)=\sum _{n=-\infty }^{N}a_{n}z^{n},}
the corresponding Hankel operator is defined as
H
f
:
C
[
z
]
→
z
−
1
C
[
[
z
−
1
]
]
.
{\displaystyle H_{f}:\mathbf {C} [z]\to \mathbf {z} ^{-1}\mathbf {C} [[z^{-1}]].}
This takes a polynomial
g
∈
C
[
z
]
{\displaystyle g\in \mathbf {C} [z]}
and sends it to the product
f
g
{\displaystyle fg}
, but discards all powers of
z
{\displaystyle z}
with a non-negative exponent, so as to give an element in
z
−
1
C
[
[
z
−
1
]
]
{\displaystyle z^{-1}\mathbf {C} [[z^{-1}]]}
, the formal power series with strictly negative exponents. The map
H
f
{\displaystyle H_{f}}
is in a natural way
C
[
z
]
{\displaystyle \mathbf {C} [z]}
-linear, and its matrix with respect to the elements
1
,
z
,
z
2
,
⋯
∈
C
[
z
]
{\displaystyle 1,z,z^{2},\dots \in \mathbf {C} [z]}
and
z
−
1
,
z
−
2
,
⋯
∈
z
−
1
C
[
[
z
−
1
]
]
{\displaystyle z^{-1},z^{-2},\dots \in z^{-1}\mathbf {C} [[z^{-1}]]}
is the Hankel matrix
[
a
1
a
2
…
a
2
a
3
…
a
3
a
4
…
⋮
⋮
⋱
]
.
{\displaystyle {\begin{bmatrix}a_{1}&a_{2}&\ldots \\a_{2}&a_{3}&\ldots \\a_{3}&a_{4}&\ldots \\\vdots &\vdots &\ddots \end{bmatrix}}.}
Any Hankel matrix arises in this way. A theorem due to Kronecker says that the rank of this matrix is finite precisely if
f
{\displaystyle f}
is a rational function, that is, a fraction of two polynomials
f
(
z
)
=
p
(
z
)
q
(
z
)
.
{\displaystyle f(z)={\frac {p(z)}{q(z)}}.}
Approximations
We are often interested in approximations of the Hankel operators, possibly by low-order operators. In order to approximate the output of the operator, we can use the spectral norm (operator 2-norm) to measure the error of our approximation. This suggests singular value decomposition as a possible technique to approximate the action of the operator.
Note that the matrix
A
{\displaystyle A}
does not have to be finite. If it is infinite, traditional methods of computing individual singular vectors will not work directly. We also require that the approximation is a Hankel matrix, which can be shown with AAK theory.
Hankel matrix transform
The Hankel matrix transform, or simply Hankel transform, of a sequence
b
k
{\displaystyle b_{k}}
is the sequence of the determinants of the Hankel matrices formed from
b
k
{\displaystyle b_{k}}
. Given an integer
n
>
0
{\displaystyle n>0}
, define the corresponding
(
n
×
n
)
{\displaystyle (n\times n)}
-dimensional Hankel matrix
B
n
{\displaystyle B_{n}}
as having the matrix elements
[
B
n
]
i
,
j
=
b
i
+
j
.
{\displaystyle [B_{n}]_{i,j}=b_{i+j}.}
Then the sequence
h
n
{\displaystyle h_{n}}
given by
h
n
=
det
B
n
{\displaystyle h_{n}=\det B_{n}}
is the Hankel transform of the sequence
b
k
.
{\displaystyle b_{k}.}
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
{\displaystyle c_{n}=\sum _{k=0}^{n}{n \choose k}b_{k}}
as the binomial transform of the sequence
b
n
{\displaystyle b_{n}}
, then one has
det
B
n
=
det
C
n
.
{\displaystyle \det B_{n}=\det C_{n}.}
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 and time-frequency representation.
= Method of moments for polynomial distributions
=The method of moments applied to polynomial distributions results in a Hankel matrix that needs to be inverted in order to obtain the weight parameters of the polynomial distribution approximation.
= Positive Hankel matrices and the Hamburger moment problems
=See also
Cauchy matrix
Jacobi operator
Toeplitz matrix, an "upside down" (that is, row-reversed) Hankel matrix
Vandermonde matrix
Notes
References
Brent R.P. (1999), "Stability of fast algorithms for structured linear systems", Fast Reliable Algorithms for Matrices with Structure (editors—T. Kailath, A.H. Sayed), ch.4 (SIAM).
Fuhrmann, Paul A. (2012). A polynomial approach to linear algebra. Universitext (2 ed.). New York, NY: Springer. doi:10.1007/978-1-4614-0338-8. ISBN 978-1-4614-0337-1. Zbl 1239.15001.
Victor Y. Pan (2001). Structured matrices and polynomials: unified superfast algorithms. Birkhäuser. ISBN 0817642404.
J.R. Partington (1988). An introduction to Hankel operators. LMS Student Texts. Vol. 13. Cambridge University Press. ISBN 0-521-36791-3.
Kata Kunci Pencarian:
- Daftar matriks yang dinamakan
- Hankel matrix
- Hermann Hankel
- Catalan number
- Symmetric matrix
- Toeplitz matrix
- Persymmetric matrix
- Hilbert matrix
- List of named matrices
- H-matrix
- Moment problem
