On the Sum of the Largest Eigenvalues of a Symmetric Matrix

The sum of the largest k eigenvalues of a symmetric matrix has a well-known extremal property that was given by Fan in 1949 [Proc. Nat. Acad. Sci., 35 (1949), pp. 652–655]. A simple proof of this property, which seems to have been overlooked in the vast literature on the subject and its many generalizations, is discussed. The key step is the observation, which is neither new nor well known, that the convex hull of the set of projection matrices of rank k is the set of symmetric matrices with eigenvalues between 0 and 1 and summing to k. The connection with the well-known Birkhoff theorem on doubly stochastic matrices is also discussed. This approach provides a very convenient characterization for the subdifferential of the eigenvalue sum, described in a separate paper.