# Nonnegative, nontrivial fixed points of orthogonal projections

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## Abstract

The mathematical problem of determining nonnegative, nontrivial fixed points (if they exist) of symmetric idempotent matrices is the central theme of the dissertation. In part, the interest in the problem stems from a result due to Pyle, who established that a linear programming problem can be reformulated as such a problem. Chapter I presents results concerning the product of two symmetric idempotent matrices, e.g., the diagonalizability of such products and the characterization of their eigenvectors in terms of orthogonality conditions. These results are used to examine the convergence of an iteration scheme based on the composition of the proximity map on a linear subspace with the proximity map on an affine subspace, and culminates in a generalization of a theorem due to Afriat. Chapter II extends these results to the composition of the proximity map on a linear subspace with the proximity map on a convex subset of the first orthant in real n- dimensional Euclidean space. A complete characterization of the fixed points of such a composition is given. These fixed points can then be used to determine nonnegative, nontrivial fixed points of a symmetric idempotent matrix. Chapter III contains results about the nonnegative, nontrivial fixed points of symmetric idempotent matrices of prescribed dimension and rank. These are used in Chapter IV, which contains a finite constructive process which simplifies the problem of finding nonnegative, nontrivial fixed points of symmetric idempotents to that of finding their zero pattern.