CONSTRUCTING A HERMITIAN MATRIX FROM ITS DIAGONAL ENTRIES AND EIGENVALUES

Given two vectors a, lambda epsilon R(n), the Schur-Horn theorem state s that a majorizes lambda if and only if there exists a Hermitian matr ix H with eigenvalues lambda and diagonal entries a. While the theory is regarded as classical by now, the known proof is not constructive. To construct a Hermitian matrix from its diagonal entries and eigenval ues therefore becomes an interesting and challenging inverse eigenvalu e problem. Two algorithms for determining the matrix numerically are p roposed in this paper. The lift and projection method is an iterative method that involves an interesting application of the Wielandt-Hoffma n theorem. The projected gradient method is a continuous method that, besides its easy implementation, offers a new proof of existence becau se of its global convergence property.