HANKEL Function
generates a Hankel matrix
- HANKEL( matrix)
where
matrix is a numeric matrix or literal.
The HANKEL function generates a Hankel matrix from
a vector, or a block Hankel matrix from a matrix.
A block Hankel matrix has the property that all
matrices on the reverse diagonals are the same.
The argument matrix is an
or
matrix; the value returned is the
result.
The Hankel function uses the first
submatrix
of the argument matrix
as the blocks of the first reverse diagonal.
The second
submatrix
of the
argument matrix forms the second reverse diagonal.
The remaining reverse diagonals are formed accordingly.
After the values in the argument matrix have all been
placed, the rest of the matrix is filled in with 0.
If
is
, then the first
columns of
the returned matrix,
, are the same as
.
If
is
, then the first
rows of
are the same as
.
The HANKEL function is especially useful in time series
applications, where the covariance matrix of a set
of variables representing the present and past and
a set of variables representing the present and
future is often assumed to be a block Hankel matrix.
If
and if
is the matrix formed by the HANKEL function, then
If
and if
is the matrix formed by the HANKEL function, then
For example, consider the following IML code:
r=hankel({1 2 3 4 5});
This code produces the following output:
R 5 rows 5 cols (numeric)
1 2 3 4 5
2 3 4 5 0
3 4 5 0 0
4 5 0 0 0
5 0 0 0 0
The following statement returns the matrix
, as shown:
r=hankel({1 2 ,
3 4 ,
5 6 ,
7 8});
R 4 rows 4 cols (numeric)
1 2 5 6
3 4 7 8
5 6 0 0
7 8 0 0
The following statement returns a different matrix
, as shown:
r=hankel({1 2 3 4 ,
5 6 7 8});
R 4 rows 4 cols (numeric)
1 2 3 4
5 6 7 8
3 4 0 0
7 8 0 0