Page 64 - Statistics and Data Analysis in Geology

Statistics and Data Analysis in Geology - Chapter 3

Few mathematical procedures have received the attention given to matrix in-
version. Dozens of methods have been devised to solve sets of simultaneous'equa-
tions, and hundreds of programmed versions exist. Some are especially tailored to
deal with special types of matrices, such as those containing many zero elements
(such matrices are called sparse) or possessing certain types of symmetry. Numer-
and
ical computation packages for personal computers, such as MATHEMATICAQ
MATLAB@, contain alternative algorithms that can be used to calculate the inverse
of matrices. Some of these procedures, such as singular value decomposition (SVD),
will find approximate inverses even when exact solutions do not exist.

Determinants

Before discussing our final topic, which is eigenvalues and eigenvectors and how
they are obtained, we must examine an additional property of a square matrix called
the determinant. A determinant is a single number extracted from a square matrix
by a series of operations, and is symbolically represented by det A, IAI, or by

It is defined as the sum of n! terms of the form

where n is the number of rows (or columns) in the matrix, the subscripts il, i2, . . . , in
are equal to 1,2,. . . , n, taken in any order, and k is the number of exchanges of
two elements necessary to place the i subscripts in the order 1,2,. . . , n. Each term
contains one element from each row and each column. The process of obtaining a
determinant from a square matrix is called evaluating the determinant
We begin the process of evaluating the determinant by selecting one element
from each row of the matrix to form a term or combination of elements. The
elements in a term are selected in order from row 1,2,. . . , n, but each combination
can contain only one element from each column. For example, we might select the
combination ~12~21~33 from a 3 x 3 matrix. Note that the method of selection
places the elements in proper order according to their first, or row, subscript. The
term contains one and only one element from each row and each column. We must
find all possible combinations of elements that can be formed in this way. If a
matrix is n x n, there will be n! combinations which contain one element from each
row and column, and whose first subscripts are in the order 1,2,. . . , n.
Since the order of multiplication of a series of numbers makes no difference
and
in the product, that is, ~11~~2~33 ~22~11~133 ~33~22~11 so on, we can
=
=
rearrange our combinations without changing the result. We wish to rearrange each
combination until the second, or column, subscript of each element is in proper
numerical order. The rearranging may be performed by swapping any two adjacent
elements. As the operation is performed, we must keep track of the number of
exchanges or transpositions necessary to get the second subscript in the correct
order. If an even number of transpositions is required (te., 0, 2, 4, 6, etc.), the
product is given a positive sign. If an odd number of transpositions is necessary
(1, 3, 5, 7, etc.), the product is negative.

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