Inversion of the Leontief Matrix by Power Series
Econometrica, 18 (2):
142--154 (1950)DOI: 10.2307/1907265
Abstract
The Leontief matrices of inter-industry transactions are large, a row for each industry in a nation. It would be desirable to invert such matrices of an order of from 100 to 200. The present paper suggests using the sum of a power series to approximate the inverse of a Leontief matrix with any desired degree of accuracy. This requires many more multiplications than do such direct methods as the Gauss-Doolittle process. But the method proposed in this paper is especially well adapted to automatic computation on the new electronic machines,in which case the large number of multiplications is not serious. The main advantage of the proposed method is that it provides an upper bound to the error of any element in the estimated inverse. A short cut method is also indicated for computing the approximation when the number of terms of the power series needed to obtain the desired degree of accuracy is large.