The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices

Abstract

Properties of the sum of the q algebraically largest eigenvalues of any real symmetric matrix as a function of the diagonal entries of the matrix are derived. Such a sum is convex but not necessarily everywhere differentiable. A convergent procedure is presented for determining a minimizing point of any such sum subject to the condition that the trace of the matrix is held constant. An implementation of this procedure is described and numerical results are included.