there exists a nonsingular T The existence of the latter allows a unique growth of the components of the vectors x or Δx that transforms one into the other. The existence of T thus illustrates the existence of a unique transformation, and also the existence of a balanced rate of growth by which one vector may be transformed into the other.
The results given here may be easily applied to matrices for different regions, where the regional outputs and final demands for a period are known but where an input-output table exists only for one region and not for the other.
The nature of the transformation matrix T also shows that we may have a balanced growth rate from one matrix to the other if we choose a suitable eigenvector and eigenvalue that will help us pass from A to B. This result is due to the similarity of the A and B matrices.
Symbols used in this chapter:
|A||Input-output coefficient matrix for country P or for base period t.|
|B||Input-output coefficient matrix for country Q or for period t + r.|
Capital coefficient matrix related to the incremental production between t + r|
|T||Transformation matrix relating A and B.|
Corresponding regression coefficient matrix used to estimate the coefficients by|
means of historical series on total intermediate output x and y, respectively.
To simplify notation, we have used the convention that the symbols Ai, Aj represent the Leontief matrices I -Ai, I - Aj, respectively.