Page 58 - Statistics and Data Analysis in Geology

Statistics and Data Analysis in Geology - Chapter 3

However, the 3 x 2 matrix cannot be postmultiplied by the 5 x 3 matrix because the
number of columns (two) in the left matrix would not equal the number of rows
(five) in the right matrix.
Multiplying a matrix by its transpose results in a square, symmetric matrix
product whose size is determined by the order of multiplication. Typically, a data
array consists of n rows and m columns, where n is much larger than m. If such
an array is premultiplied by its transpose, the minor product matrix will be m x m:

But reversing the order of multiplication yields the n x n major product matrix:

The equation for the general case of matrix multiplication is

In a series of multiplications, the sequence in which the multiplications are
accomplished is not mandatory if the arrangement is not changed. That is,

A x B X C = (A X B) X C = A X (B X C)
Because powering is simply a series of multiplications, a square matrix can be
raised to a power. So,
A~=AXA
and
A3 = A2 x A = A X A X A

Note that nonsquare matrices cannot be powered, because the number of rows and
columns of a rectangular matrix would not accord if the matrix were multiplied by
itself.
As an example, we can power the array of transition probabilities discussed at
the first of this section. In matrix form,
0.74 0.23 0.03

0.05 0.38 0.57

0.572 0.322 0.106 I
0.150 0.505 0.345
0.104 0.460 0.437

and
0.461 0.368 0.171 I
0.178 0.474 0.348
0.144 0.470 0.385

130