Page 77 - Statistics and Data Analysis in Geology

Matrix Algebra

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Figure 3-2. Elongated ellipse defined by rows in matrix of correlations between chromium
and vanadium, which are highly correlated.

: - o ’ 2

-2.0
-2.0 -1 .o .O 1 .o 2 0
MG
Figure 3-3. Nearly circular ellipse defined by rows in matrix of correlations between mag-
nesium and vanadium, which have a correlation approaching zero.

The two eigenvalues of this matrix are

hi = 1.029 A2 = 0.971

which are almost identical in size. As we can see, they define the major and minor
semiaxes of an ellipse that is almost a circle, and both the semiaxes and the vectors
are essentially radii. By definition, the axes of the ellipse are at right angles to each
other, and the two plotted vectors also are almost orthogonal.
Some final notes on eigenvalues: You’ll notice that the correlation matrices
we’ve graphed are square, symmetrical about their diagonals, composed of real
elements (that is, no imaginary numbers), and that the largest numbers in every row

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