The Mathematics of Financial Modelingand Investment Management

5-Matrix Algebra Page 153 Wednesday, February 4, 2004 12:49 PM

Matrix Algebra 153

H 0 H 1 H 2 ...

H 1 H 2 ......

H = H 2 .........

...

...

The rank of a Hankel matrix can be defined in three different ways:

1. The column rank is the largest number of linearly independent
   sequence columns.


2. The row rank is the largest number of linearly independent sequence
   rows.


3. The rank is the superior of the ranks of all finite matrices of the type:

H 0 H 1 · HN'

H 1 H 2 · ·

HNN, ' =

· · · ·

HN · · HNN+ '

As in the finite-dimensional case, the three definitions are equivalent in

the sense that the three numbers are equal, if finite, or they are all three

infinite.

VECTOR AND MATRIX OPERATIONS

Let’s now introduce the most common operations performed on vectors

and matrices. An operation is a mapping that operates on scalars, vectors,

and matrices to produce new scalars, vectors, or matrices. The notion of

operations performed on a set of objects to produce another object of the

same set is the key concept of algebra. Let’s start with vector operations.

Vector Operations

The following operations are usually defined on vectors: (1) transpose,

(2) addition, and (3) multiplication.

Transpose

The transpose operation transforms a row vector into a column vector and

vice versa. Given the row vector x = [x 1 ...xn] its transpose, denoted as xT

or x′, is the column vector: