Minimization of norms and the spectral radius of a sum of nonnegative matrices under diagonal equivalence
The authors generalize in various directions a result of \textit{S. Friedland} and \textit{S. Karlin} [Duke Math. J. 42, 459-490 (1975; Zbl 0373.15008)] on a lower bound for the spectral radius of a matrix that is positively diagonally equivalent to a doubly stochastic matrix. First, they complete the original result by characterizing the equality cases for doubly stochastic matrices of general zero-pattern. Further, they discuss the sums of matrices that are diagonally equivalent to doubly stochastic matrices. After introducing a lower bound for submultiplicative norms of such matrices and characterizing those cases in which the lower bound is attained, these results are used to obtain a lower bound for the spectral radius of such sums and to characterize the equality case for the spectral radius. Finally, they introduce a lower bound for the submultiplicative norms of sums of nonnegative matrices in terms of norms of certain generalized doubly stochastic matrices, characterize those cases in which the lower bound is attained, and also obtain a lower bound for the spectral radius of such sums and characterize the corresponding equality case.
scientific article; zbMATH DE number 931898
scientific article