The Mathematics of Financial Modelingand Investment Management

5-Matrix Algebra Page 152 Wednesday, February 4, 2004 12:49 PM

152 The Mathematics of Financial Modeling and Investment Management

HANKEL MATRIX

For the theoretical analysis of the autoregressive integrated moving

averages (ARMA) processes described in Chapter 11, it is important to

understand a special type of matrix, a Hankel matrix. A Hankel matrix

is a matrix where for each antidiagonal the element is the same. For

example, consider the following square Hankel matrix:

17 16 15 24

16 15 24 33

15 24 33 72

24 33 72 41

Each antidiagonal has the same value. Now consider the elements of the

antidiagonal running from the second row, first column and first row,

second column. Both elements have the value 16. Consider another

antidiagonal running from the fourth row, second column to the second

row, fourth column. All of the elements have the value 33.

An example of a rectangular Hankel matrix would be

72 60 55 43 30 21

60 55 43 30 21 10

55 43 30 21 10 80

Notice that a Hankel matrix is a symmetric matrix.^3

Consider an infinite sequence of square n×n matrices:

H 0 , H 1 , ..., Hi, ...

The infinite Hankel matrix H is the following matrix:

(^3) A special case of a Hankel matrix is when the values for the elements in the first
row of the matrix are repeated in each successive row such that its value appears one
column to the left. For example, consider the following square Hankel matrix:
41 32 23 14
32 23 14 41
23 14 41 32
14 41 32 23
This type of Hankel matrix is called an anticirculant matrix.