Each of the three arrays in (4.2) or (4.4) constitutes a matrix.

A matrix is defined as a rectangular array of numbers, parameters, or variables. The members of the array, referred to as the elements of the matrix, are usually enclosed in brackets, as in (4.2), or sometimes in parentheses or with double vertical lines: || ||. Note that in matrix A (the coefficient matrix of the equation system), the elements are separated not by commas but by blank spaces only. As a shorthand device, the array in matrix A can be written more simply as

Inasmuch as the location of each element in a matrix is unequivocally fixed by the subscript, every matrix is an ordered set.

Vectors as Special Matrices

The number of rows and the number of columns in a matrix together define the dimension of the matrix. Since matrix A in (4.2) contains m rows and n columns, it is said to be of dimension m X n (read: "m by n"). It is important to remember that the row number always precedes the column number; this is in line with the way the two subscripts in atj are ordered. In the special case where m = n, the matrix jis called a square matrix-, thus the matrix A in (4.4) is a 3 X 3 square matrix.

Some matrices may contain only one column, such as x and d in (4.2) or (4.4). Such matrices are given the special name column vectors. In (4.2), the dimension of x is n X 1, and that of d is m X 1; in (4.4) both x and d are 3 X 1. If we arranged the variables Xj in a horizontal array, though, there would result a 1 X n matrix, which is called a row vector. For notation purposes, a row vector is often distinguished from a column vector by the use of a primed symbol:

You may observe that a vector (whether row or column) is merely an ordered «-tuple, and as such it may be interpreted as a point in an «-dimensional space. In turn, the m X n matrix A can be interpreted as an ordered set of m row vectors or as an ordered set of n column vectors. These ideas will be followed up later.

An issue of more immediate interest is how the matrix notation can enable us, as promised, to express an equation system in a compact way. With the matrices defined in (4.4), we can express the equation system (4.3) simply as

In fact, if A, x, and d are given the meanings in (4.2), then even the general-equation system in (4.1) can be written as Ax — d. The compactness of this notation jg thus unmistakable.

However, the equation Ax = d prompts at least two questions. How do we multiply two matrices A and x? What is meant by the equality of Ax and dl Since matrices involve whole blocks of numbers, the familiar algebraic operations defined for single numbers are not directly applicable, and there is need for a new set of operational rules.

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