Page 78 - Statistics and Data Analysis in Geology

Statistics and Data Analysis in Geology - Chapter 3

are on the diagonal. As a consequence of these special conditions, the eigenvalues
will always be real numbers that are equal to or greater than zero. As you can
verify by checlung these examples, the sum of the eigenvalues of a matrix is always
equal to the sum of the diagonal elements, or the trace, of the original matrix.
In a correlation matrix, the diagonal elements are all equal to one, so the trace is
simply the number of variables. The product of the eigenvalues will be equal to the
determinant of the original matrix. Most (but not all) of the eigenvalue operations
we will consider later will be applied to correlation or covariance matrices, so these
special results will hold true in most instances. The methods just developed can be
extended directly to n x n matrices, although the procedure becomes increasingly
cumbersome with larger matrices.

E igenvect ors

We can examine the correlation matrices we calculated for the Istrian vineyard data
to gain some insight into the geometrical nature of eigenvectors. First, consider the
2 x 2 matrix

with eigenvalues
A1 = 1.28 A2 = 0.72
Substituting the first eigenvalue into the original matrix gives

1 - 1.28 -0.28 ] = [ -0.28 -0.28 1
-0.28 1 - 1.28 -0.28 -0.28

whose solution is the eigenvector
[ 4 = [ -:]

In Figure 3-1, we can interpret this eigenvector as the slope of the major semi-
axis of the enclosing ellipse. If we regard the elements of the eigenvector as coor-
dinates, the first eigenvector defines an axis whch extends from the center of the
ellipse into the second quadrant at an angle of 135". The length is equal to the first
eigenvalue, or 1.28.
Turning to the second eigenvalue, A2 = 0.72, the equation set is

1 - 0.72 -0.28 ] = [ 0.28 -0.28 1
-0.28 1 - 0.72 -0.28 0.28

whose solution gives the second eigenvector:
[::I = [ :]

In Figure 3-1, ths will plot as the vector drection l/l = 45", perpendicular to the
major semiaxis of the ellipse. Its magnitude or length is 0.72.

150