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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.
Item Metadata
Title |
Theory and applications of compound matrices
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
1956
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Description |
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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Genre | |
Type | |
Language |
eng
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Date Available |
2012-02-09
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Provider |
Vancouver : University of British Columbia Library
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Rights |
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.
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DOI |
10.14288/1.0080655
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Campus | |
Scholarly Level |
Graduate
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Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.