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Theory and applications of compound matrices Thompson, Robert Charles

Abstract

If A is an n-square matrix, the p-th compound of A is a matrix whose entries are the p-th order minors of A arranged in a doubly lexicographic order . In this thesis some of the theory of compound matrices is given, including a short proof of the Sylvester-Franke theorem. This theory is used to obtain an extremum property of elementary symmetric functions of the k largest (or smallest) eigenvalues of non-negative Hermitian (n.n.h) matrices. Extensions of theorems due to Weyl and Wielandt are given. The first of these relates elementary symmetric functions of singular values of any matrix A with the same elementary symmetric functions of the eigenvalues. The second gives an extremum property of arbitrary eigenvalues of n.n.h matrices and enables inequalities relating the eigenvalues of A, B with the eigenvalues of A + B to be given (A, B, n.n.h.). Finally, a norm inequality for an arbitrary matrix is given.

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