Further information is available if all the elements of M are positive, or if none is negative with an assurance of purely positive components for the eigenvector; then the minimum component can be used to reverse the inequality and reveal the least row sum as a lower bound to the eigenvalue. So it will be for the largest eigenvector of an irreducible nonnegative matrix, according to the classical theory of Perron and Frobenius .
Statistical information concerning the eigenvalue can be obtained from the equation just as easily as the traditional bounds; for example, suppose that the equations for each component are summed:
With the definitions
provided that the sum of the eigenvector components is not zero; again it is a foregone conclusion that the eigenvector of the largest eigenvalue of a nonnegative matrix can be so chosen.
Inasmuch as the right hand side has the form of an inner product, as long as the quantities involved are real, there is an angle and vectors and for which
Even if complex quantities were involved, there would only be a lost phase, which could be restored. In such an equation is circumscribed by the variances of the set of column sums, and of the eigenvector components.
Harold V. McIntosh