Page 69 - Statistics and Data Analysis in Geology

Matrix Algebra

E igenva I u es a n d E igenvect ors
The topic we will consider next usually is regarded as one of the most difficult topics
in matrix algebra, the determination of eigenvalues and eigenvectors (also called
“latent” and “proper” values and vectors). The difficulty is not in their calculation,
which is cumbersome but no more so than many other mathematical procedures.
Rather, difficulties arise in developing a “feel” for the meaning of these quantities,
especially in an intuitive sense. Unfortunately, many textbooks provide no help in
this regard, placing their discussions in strictly mathematical terms that may be
difficult for nonmathematicians to interpret.
A lucid discussion and geometric interpretation of eigenvectors and eigenval-
ues was prepared by Peter Gould for the benefit of geography students at Pennsyl-
vania State University. The following discussion leans heavily on his prepared notes
and a subsequent article (Gould, 1967). We will consider a real matrix of coordi-
nates of points in space and interpret the eigenvalues and associated functions as
geometric properties of the arrangement of these points. This approach limits us,
of course, to small matrices, but the insights gained can be extrapolated to larger
systems even though hand computation becomes impractical. In this regard, it may
be noted that we are entering a realm where the computational powers of even the
largest computers may be inadequate to solve real problems.
Eigenva I ues

Having worked through determinants, we can use them to develop eigenvalues.
Consider a hypothetical set of simultaneous equations expressed in the following
matrix form:
AX = AX (3.4)
This equation states that the matrix of coefficients (the Uij’S) times the vector of
unknowns (the xi’s) is equal to some constant (A) times the unknown vector itself.
The problem is the same as in the solution of the simultaneous equation set
AX=B

except now
B=hX
Our concern is to find values of h that satisfy this relationship. Equation (3.4)
can be rewritten in the form
(A - h I) X = 0 (3.5)
where h I is nothing more than an identity matrix (of the same size as A) times the
[: : :]
quantity A. That is,

hI= 0 h 0

for a 3 x 3 matrix. Written in conventional form, the equivalent of the three simul-
taneous equations is

(all - h) x1 + d12x2 + d.13x3 = 0

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