Rank-one corrections of nonnegative matrices, with an application to matrix population models

SIAM Journal on Matrix Analysis and Applications. 2014. Vol. 35. No. 2. P. 749–764.

Protasov V., Logofet D. O.

We study the location of λ2(A), the second positive eigenvalue of a nonnegative matrix A, as the issue of how many positive eigenvalues can be shifted beyond the spectral radius ρ(A) by means of arbitrary changes in elements of one row. The notion of rank-one correction suggests the nearest generalization expanding the changes in one row to any matrix of rank one (still keeping the matrix nonnegative). The main theorem limits the number of those eigenvalues, counting multiplicities, to the increased spectral radius alone. In matrix population models, we treat the projection matrix L = T + F as the rank-one correction of its transition part T by the fertility one F. The matrix T is column substochastic due to its demographic interpretation, hence we conclude that λ2(L) ≤ 1 and specify the rare cases where λ2(L) = 1. The location λ2(L) < 1 ensures that the function R(L) = 1 − det (I − L) has the indicator property, namely, its value is always located on the same side of 1 as is ρ(L). This indicator does not pose any computational problems and helps calibrate L from empirical data.

Keywords: indicator, population dynamics, nonnegative matrix, Perron Frobenius theorem, leading eigenvalue, rank-one correction, projection matrices