Unconstrained Variational Principles for Eigenvalues of Real Symmetric Matrices

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Abstract
Certain real-valued functions, whose critical points and critical values are related to the eigenvalues and eigenvectors of a real symmetric matrix, are described and analyzed. These functions, in general, are smooth and bounded below. Variational principles for finding various specific eigenvalues and eigenvectors of the matrix A are described. These problems have a Morse theory. They may be written as the difference of two convex functions, so there are also natural dual problems that include the classical constrained variational principles for eigenvalues and eigenvectors.

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