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 um by «"). 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 aSJ are ordered. In the special case where m = h, the matrix is 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 I, and that of d is m x 1; in (4.4) both x and d are 3 x 1. If we arranged the variables xy 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 w-tuple, and as such it may sometimes 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 in Chap. 5.

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 die matrices defined in (4.4). we can express the equation system (4.3) simply as x - [-ï] x2 ■ ■■ xn]

In fact, if A, 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 is thus unmistakable.

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

EXERCISE 4.1

1. Rewrite the market model (3.1) in the format of (4.1), and $how that, if the three variables are arranged in the order QSt and Pt the coefficient matrix wili be

How would you write the vector of constants?

2. Rewrite the market model (3.12) in the format of (4.1) with the variables arranged in the following order: Qdh QsU QdZf P}f P2. Write out the coefficient matrix, the variable vector, and the constant vector

3. Can the market model (3,6) be rewritten in the format of (4.1)? Why?

4. Rewrite the nationai-income model (3.23) in the format of (4.1), with V as the first variable. Write out the coefficient matrix and the constant vector.

5. Rewrite the national ncome model of Exercise 3.5-1 in the format of (4,1), with the variables in the order V, 7", and G [Hint: Watch out for the multiplicative expression b(V - T) in the consumption function.]

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