On the Sum of the Largest Eigenvalues of a Symmetric Matrix
Copyright policy )On the Sum of the Largest Eigenvalues of a Symmetric Matrix
The sum of the largest \(k\) eigenvalues of a symmetric matrix has an extremal property that was given by \textit{Ky Fan} [Proc. Natl. Acad. Sci. USA 35, 652--655 (1949; Zbl 0041.00602)]. A simple proof of this property is discussed in this paper. The key step of this proof is based on the observation 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 Birkhoff theorem on doubly stochastic matrices is also discussed.
extremal property, symmetric matrix, Miscellaneous inequalities involving matrices, sum of the largest eigenvalues, Norms of matrices, numerical range, applications of functional analysis to matrix theory, Birkhoff theorem, Inequalities involving eigenvalues and eigenvectors, convex hull, projection matrices, doubly stochastic matrices, Stochastic matrices
extremal property, symmetric matrix, Miscellaneous inequalities involving matrices, sum of the largest eigenvalues, Norms of matrices, numerical range, applications of functional analysis to matrix theory, Birkhoff theorem, Inequalities involving eigenvalues and eigenvectors, convex hull, projection matrices, doubly stochastic matrices, Stochastic matrices
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