Page 73 - Statistics and Data Analysis in Geology

Matrix Algebra

These are complex numbers, containing both real parts and imaginary parts which
include the imaginary number, i = a. Fortunately, a symmetric matrix always
yields real eigenvalues, and most of our computations involving eigenvalues and
eigenvectors will utilize covariance, correlation, or similarity matrices which are
always symmetrical.
Next, we will consider the eigenvalues of the third-order matrix:

20 -4
[ -40 8 -2:]
-60 12 -26

20-h -4 8
-40 8-h -20 =O
-60 12 -26-h

Expanding out the determinant and combining terms yields

-A3 + 2h2 + 8h = 0

This is a cubic equation having three roots that must be found. In this instance,
the polynomial can be factored into

(A - 4) (A - 0) (A + 2) = 0

and the roots are directly obtainable:

h1=+4 h2=O &=-2

Although the techniques we have been using are extendible to any size matrix,
finding the roots of large polynomial equations can be an arduous task. Usually,
eigenvalues are not found by solution of a polynomial equation, but rather by ma-
trix manipulation methods that involve refinement of a successive series of approx-
imations to the eigenvalues. These methods are practical only because of the great
computational speed of digital computers. Utilizing this speed, a researcher can
compress literally a lifetime of trial solutions and refinements into a few minutes.
We can now define another measure of the “size” of a square matrix. The rank
of a square matrix is the number of independent rows (or columns) in the matrix
and is equal to the number of nonzero eigenvalues that can be extracted from the
matrix. A nonsingular matrix has as many nonzero eigenvalues as there are rows
or columns in the matrix, so its rank is equal to its order. A singular matrix has
one or more rows or columns that are dependent on other rows or columns, and
consequently will have one or more zero eigenvalues; its rank will be less than its
order.
Now that we have an idea of the manipulations that produce eigenvalues, we
may try to get some insight into their nature. The rows of a matrix can be regarded
as the coordinates of points in m-dimensional space. If we restrict our considera-
tion to 2 x 2 matrices, we can represent this space as an illustration on a page and
can view matrix operations geometrically.

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