Constructing a Hermitian Matrix from Its Diagonal Entries and Eigenvalues

Given two vectors a; 2 Rn, the Schur-Horn theorem states that a majorizes if and only if there exists a Hermitian matrix H with eigenvalues 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 eigenvalues therefore becomes an interesting and challenging inverse eigenvalue problem. Two algorithms for determining the matrix numerically are proposed in this paper. The lift and projection method is an iterative method which involves an interesting application of the Wielandt-Ho man theorem. The projected gradient method is a continuous method which, besides its easy implementation, o ers a new proof of existence because of its global convergence property.