On the Mathematical Transformation of Input-Output Matrices over Time or Space
|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.