DOMINANT EIGENVALUES UNDER TRACE-PRESERVING DIAGONAL PERTURBATIONS

For an essentially nonnegative matrix A, we consider the problem of mi nimizing the dominant eigenvalue of A + D over all real diagonal matri ces D with zero trace. The solution is closely related to the unique l ine-sum-symmetric diagonal similarity of A in the irreducible case, an d 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 a n explicit function of the entries of A in some cases, and in terms of the zeros of polynomials with coefficients constructed from the entri es of A in others.