Page 51 - Statistics and Data Analysis in Geology

ChaDter 3

Matrix Ggebra

This chapter is devoted to matrix algebra. Most of the methods we will dis-
cuss in subsequent chapters are based on matrix manipulations, especially as per-
formed by computers. In this chapter, we will examine the mathematical operations
that underlie such techniques as trend-surface analysis, principal components, and
discriminant functions. These techniques are almost impossible to apply without
the help of computers, because the calculations are complicated and must be per-
formed repetitively. However, with matrix algebra we can express the basic princi-
ples involved in a manner that is succinct and easily understood. Once you master
the rudiments of matrix algebra, you will be able to see the fundamental structure
within the complex procedures we will examine later.
Most geologists probably have not taken a course in matrix algebra. This is un-
fortunate; the subject is not difficult and is probably one of the most useful tools in
mathematics. College courses in matrix algebra usually are sprinkled liberally with
theorems and their proofs. Such an approach is certainly beyond the scope of this
short chapter, so we will confine ourselves to those topics pertinent to techniques
that we will utilize later. Rather than giving derivations and proofs, the material
will be presented by examples.

The Matrix

A matrix is a rectangular array of numbers, exactly the same as a table of data. In
matrix algebra, the array is considered to be a single entity rather than a collection
of individual values and is operated upon as a unit. This results in a great simpli-
fication of the statement of complicated procedures and relationships. Individual
numbers within a matrix are called the elements of the matrix and are identified
by subscripts. The first subscript specifies the row in which the element occurs
and the second specifies the column. The individual elements of a matrix may be