Page 52 - Statistics and Data Analysis in Geology

Statistics and Data Analysis in Geology - Chapter 3

measurements of variables, variances or covariances, sums of observations, terms
in a series of simultaneous equations or, in fact, any set of numbers.
As an example, in Chapter 2 you were asked to compute the variances and
covariances of trace-element data given in Table 2-3. Your answers can be arranged
in the form of the matrix below.

We can designate a matrix (perhaps containing values of several variables) sym-
bolically by capital letters such as [XI, XI (X), or IlXll. In a change from earlier edi-
tions of this book, we will adopt the commonly used boldface notation for matrices.
Individual entries in a matrix, or its elements, are indicated by subscripted italic
lowercase letters such as Xij. Particularly in older books, you may encounter dif-
ferent conventions for denoting individual elements of a matrix. The symbol xij is
the element in the ith row and the jth column of matrix X. For example, if X is the
x=[i i]
3 x 3 matrix

is
x33 is 9, ~ 1 3 7, x21 is 2, and so on. The order of a matrix is an expression
of its size, in the sense of the number of rows and/or the number of columns it
contains. So, the order of X, above, is 3. If the number of rows equals the number
of columns, the matrix is square. Entries in a square matrix whose subscripts are
equal (ie., i = j) are called the diagonal elements, and they lie on the principal
diagonal or major diagonal of the matrix. In the matrix of trace-element variances
and covariances, the variances lie on the diagonal and the off-diagonal elements
are the covariances. The diagonal elements in the matrix above are 1, 5, and 9.
Although data arrays usually are in the form of rectangular matrices, often we will
create square matrices from them by calculating their variances and covariances
or other summary statistics. Many useful operations that can be performed on
square matrices are not possible with nonsquare matrices. However, two forms
of nonsquare matrices are especially important; these are the vectors, 1 x m (row
vector) and m x 1 (column vector).
Certain square matrices have special importance and are designated by name.
A symmetric matrix is a square matrix in which all observations Xij = Xji, as for
[: : '1
example

3 5 6

The variance-covariance matrix of trace elements given above is another example
of a square matrix that is symmetrical about the diagonal.
A diagonal matrix is a square, symmetric matrix in which all the off-diagonal
elements are 0. If all of the diagonal elements of a diagonal matrix are equal, the
matrix is a scalar matrix. Finally, a scalar matrix whose diagonal elements are equal
to 1 is called an identity matrix or unit matrix. An identity matrix is almost always

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