Minimization of norms and the spectral radius of a sum of nonnegative matrices under diagonal equivalence Original Research Article

We generalize in various directions a result of Friedland and Karlin on a lower bound for the spectral radius of a matrix that is positively diagonally equivalent to a doubly stochastic matrix. The original result characterizes the equality case for two special zero patterns of the doubly stochastic matrix. Here we characterize the equality cases for doubly stochastic matrices of general zero pattern. We further generalize the results to sums of matrices that are diagonally equivalent to doubly stochastic matrices. Our claims follow from inequalities we prove on norms of matrices. Finally, we prove the corresponding inequalities (and equalities) for nonnegative matrices that are not sums of matrices diagonally equivalent to doubly stochastic matrices.