The Second Eigenvalue of the Leontief Matrix
András Bródy · Economic Systems Research · 1997
According to Frobenius, a positive matrix possesses a unique positive eigenvector which belongs to a positive eigenvalue. This eigenvalue is of the largest absolute magnitude and the matrix admits no other positive eigenvector. If an arbitrary positive vector is repeatedly premultiplied by such a matrix, then the result tends towards this positive eigenvector. It is the second largest eigenvalue that determines the speed of convergence. The estimate of the second eigenvalue of a purely random flow coefficient matrix shows that its expected absolute magnitude declines monotonically with the size of the matrix. Hence, the larger the system is the faster is the convergence. A prescribed exactness of the eigenvector (of equilibrium prices or quantities) will be reached after a few—perhaps just a couple of—iterations in a large system.