5-Matrix Algebra Page 141 Wednesday, February 4, 2004 12:49 PM

CHAPTER

5

Matrix Algebra

O

rdinary algebra deals with operations such as addition and multiplica-

tion performed on individual numbers. In many applications, however,

it is useful to consider operations performed on ordered arrays of num-

bers. This is the domain of matrix algebra. Ordered arrays of numbers are

called vectors and matrices while individual numbers are called scalars. In

this chapter, we will discuss the basic operations of matrix algebra.

VECTORS AND MATRICES DEFINED

Let’s now define precisely the concepts of vector and matrix. Though

vectors can be thought of as particular matrices, in many cases it is use-

ful to keep the two concepts—vectors and matrices—distinct. In partic-

ular, a number of important concepts and properties can be defined for

vectors but do not generalize easily to matrices.^1

Vectors

An n-dimensional vector is an ordered array of n numbers. Vectors are

generally indicated with bold-face lower case letters. Thus a vector x is

an array of the form

x = [x 1 ...xn ]

The numbers xi are called the components of the vector x.

A vector is identified by the set of its components. Consider the vec-

tors x = [x 1 ...xn] and y = [y 1 ...ym]. Two vectors are said to be equal if

(^1) Vectors can be thought as the elements of an abstract linear space while matrices
are operators that operate on linear spaces.
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