Minimum symbol error probability MIMO design under the per-antenna power constraint
Journal of Electrical and Computer Engineering
Constructing all self-adjoint matrices with prescribed spectrum and diagonal
Advances in Computational Mathematics
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Given two vectors $a,\lambda \in R^{n}$, the Schur--Horn theorem states that $a$ majorizes $\lambda$ if and only if there exists a Hermitian matrix $H$ 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 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 that involves an interesting application of the Wielandt--Hoffman theorem. The projected gradient method is a continuous method that, besides its easy implementation, offers a new proof of existence because of its global convergence property.