Bounds for Levinger’s function of nonnegative almost skew-symmetric matrices

The analysis of the Perron eigenspace of a nonnegative matrix A whose symmetric part has rank one is continued. Improved bounds for the Perron root of Levinger’s transformation (1 − α)A + αAt (α ∈ [0, 1]) and its derivative are obtained. The relative geometry of the corresponding left and right Perron vectors is examined. The results are applied to tournament matrices to obtain a comparison result for their spectral radii.