Page 56 - Statistics and Data Analysis in Geology

Statistics and Data Analysis in Geology - Chapter 3

can determine the probability of attaining specified states at successive intervals
by powering the transition probability matrix. That is, the probability matrix, P,
after n steps through the succession is equal to Pn. The nth power of a matrix is
simply the matrix times itself n times. To perform this operation, however, we
must know the special procedures of matrix multiplication.
The simplest form of multiplication involves two square matrices, A and B, of
equal size, producing the product matrix, C. An easy method of performing this
operation is to arrange the matrices in the following manner:

To obtain the value of an element Cij, multiply each element of row i of A, starting
at the left, by each element of column j of B, starting at the top. All the products
are summed to obtain the Cij element of the answer. The steps in multiplication
are demonstrated below on the two matrices,

First, multiply a11 by bll = 1,

Then, a12 by b21 = 12,

: t]
f
Finally, 6.13 by b31 = 35,
0 6 7

The entry cll is the sum of these three values, 1 + 12 + 35 = 48. These steps can be
summarized in the diagram below. Note that each entry Cij in the product matrix
results from multiplying and summing the products of elements in the ith row of
matrix A by elements in the jth column of matrix B.

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