Page 79 - Statistics and Data Analysis in Geology

Matrix Algebra

We can determine the eigenvalues for the matrix of correlations between chro-
mi= and vanadium in a similar fashion. The matrix is

with eigenvalues
hi = 1.85 A2 = 0.15

The first eigenvector is

1 - 1.85 0.85 1 = 1-0.85 0.85 1
1 0.85 1 - 1.85 0.85 -0.85
L A L A
[::I = [:I

which defines a line having a slope of 45". This axis bisects the angle between the
two points and the center of the ellipse in Figure 3-2. The magnitude of the major
semiaxis is equal to 1.85, the first eigenvalue of RC7,,,. Similarly, we can show that
the eigenvector associated with th( second eigenvalue is

1-0.15 0.85 ] = [ 0.85 0.851
0.85 1 - 0.15 0.85 0.85
::I
[ = [-:I

This procedure can be applied to the matrix Rmg,,, and the eigenvectors found
will again define directions of 135" and 45", as shown in Figure 3-3. By now you
no doubt suspect that the eigenvectors of 2 x 2 symmetric matrices will always
lie at these specific angles, and this is indeed the case. The eigenvectors of real,
symmetric matrices are always orthogonal, or at right angles to each other. This is
not true of eigenvectors of matrices in general, but only of symmetric matrices. In
addition, the eigenvectors of two-dimensional symmetric matrices are additionally
constrained to orientations that are multiples of 45". Incidentally, if two vectors,
A and B, are orthogonal, then ATB = 0.
Eigenvalue and eigenvector techniques are directly extendible to larger matri-
ces, even though the operations become tedious. As an example, we will consider
the full 5 x 5 correlation matrix R for trace metals from Istrian vineyard soils. The
five eigenvalues of this matrix are
A= 12.453 1.233 0.789 0.465 0.061 ]
L
and their associated eigenvectors are

0.585 -0.248 0.259 1::!::] [ -0.727 0.062
-0.363 -0.075 0.95 1
Vp = [ 0.736 v4 = -0.628 Vs = -0.023
0.498 -0.490 0.052 -0.398 0.593
0.469 0.389 0.300 0.652 0.339

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