# Advances in Input-Output Analysis: Technology, Planning, and Development

By William Peterson | Go to book overview

2
On the Mathematical Transformation of Input-Output Matrices over Time or Space
A. GHOSH
THE TRANSFORMATION RULE
One feature of input-output matrices is that if they are of the same order and have the same classification it may be possible to find a transformation between any two of them. If such a transformation can be related to the output vectors concerned by a scalar or a diagonal matrix, then the resulting transformation of the matrices can easily be computed and can be expressed in canonical form, and the implicit technological transformation that lies behind this transformation can be brought out clearly.The object of this chapter is to demonstrate some useful properties of input- output matrices of the same order, conceptualized as forming a group.A group is defined as a set of elements, and a method of combining them, satisfying the following conditions.Let (Ai), the typical input-output matrix,1 be a member of the group.
 1 Then if (Ai) and (Aj) are members of the group, the product (Ai)(Aj) and (Aj)(Ai) are also members of the group, that is, the product sets also satisfy the usual conditions on the coefficient matrix. 2 [(Ai·Aj)(Ak)]= [Ai(Aj · Ak)] 3 There is an element I such that I · Ai = Ai · I and for every (Ai) we have (Ai · Ai-1) = I.

A subgroup is any subset of elements made up of the members of a group that itself satisfies the definition of a group. A right or left co-set of a subgroup g or a group G is the set of elements of G obtained by multiplying each of the elements of g in turn (using right or left multiplication, respectively) by some element of G not in g. Using the above properties we can conceptually formulate that the elements (Ai) are members of the group G.

Under the conventional restrictions on the input-output coefficients, the existence of the Leontief inverse has been proved by various authors. The associative law also follows from the property of matrix multiplication itself.

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