The trend seems to be favorable; generally the average value approximates the nominal value, but there are distortions at both ends of the range which can be attributed to the influence of the bounds to which the tensor squares are subject, but which would not affect a more random collection of matrices.

In order to have in Eq. 25 it is necessary that , , or . In the first case all the column sums will be equal, equal to the eigenvalue, and the matrix will be stochastic. In the second case the vector will be an eigenvector, the row sums will all be constant, and again the matrix will be stochastic. The final alternative is more interesting, requiring the residuals in the column sums to be orthogonal to the residuals in the eigenvector, for a matrix which is explicitly not stochastic.

Harold V. McIntosh