Minimization of Norms and the Spectral Radius of a Sum of Nonnegative Matrices Under Diagonal Equivalence

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 • The research of these authors was supported by their joint grant No. 90-00434 from the United States-Israel Binational Science Foundation, Jerusalem, Israel. t The research of this author was supported in part by NSF Grant DMS-9306357. t The research of this author was supported in part by NSF Grant DMS-9123318. LINEAR ALGEBRA AND ITS APPLICATIONS 241-243:431-453 (1996) © Elsevier Science Inc., 1996 655 Avenue of the Americas, New York, NY 10010 0024-3795/96/$15.00 SSDI 0024-3795(95)OO521-R 432 D. HERSHKo\VITZ ET AL. 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 slims 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.