Courant-Fischer and Graph Coloring
نویسنده
چکیده
4.1 Eigenvalues and Optimization I cannot believe that I have managed to teach three lectures on spectral graph theory without giving the characterization of eigenvalues as solutions to optimization problems. It is one of the most useful ways of understanding eigenvalues of symmetric matrices. To begin, let A be a symmetric matrix with eigenvalues α1 ≥ α2 ≥ · · · ≥ αn, and corresponding orthonormal eigenvectors v1, . . . , vn. Lemma 4.1.1. α1 = max x xTAx xTx .
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