Page 59 - Statistics and Data Analysis in Geology

Matrix Algebra

If we continue to power the transition probability matrix, it converges to a sta-
ble configuration (called the stationary probability matrix) in which each column
of the matrix is a constant. These are the proportions of the specific lithologies
represented by the columns. In this example, the proportions are 23% sandstone,
45% shale, and 32% limestone. We can see that the columns are converging on these
values at the 10th power of T:

0.248 0.443 0.309
0.230 0.449 0.321 1
0.228 0.450 0.322

Square matrices also can be raised to a fractional power, most commonly to
the one-half power. This is equivalent to finding the square root of the matrix. That
is, All2 is a matrix, XI whose square is A:

Finding fractional powers of matrices can be computationally troublesome.
Fortunately, in the applications we will consider, we will only need to find the frac-
tional powers of diagonal matrices, which have special properties that make it easy
to raise them to a fractional power. If we raise the diagonal matrix A to the one-
half power, the result is a diagonal matrix whose nonzero elements are equal to the
square roots of the equivalent elements in A. For example, if A is 3 x 3,

As we defined it earlier, the identity matrix is a special diagonal matrix in which
the diagonal terms are all equal to 1. The identity matrix has an extremely useful
property; if a matrix is multiplied by an identity matrix, the resulting product is
exactly the same as the initial matrix:

100 147
258XO10=258
[: :] [O 0 11 [3 6 91

Thus, the identity matrix corresponds to the 1 of ordinary multiplication. This
property is especially important in operations in the following sections.

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