Abstract
For an essentially nonnegative matrix A, we consider the problem of minimizing the dominant eigenvalue of A+ D over real diagonal matrices D with zero trace. The solution is closely related to the unique line-sum-symmetric diagonal similarity of A in the irreducible case, and we describe the solution for general essentially nonnegative A. The minimizer D is always unique, and we characterize those matrices A for which the minimizer D is 0. We solve the problem for several classes of matrices by finding the line-sum-symmetric diagonal similarity as an explicit function of the entries of A in some cases, and in terms of the zeros of polynomials with coefficients constructed from the entries of A in others.
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