Year of Publication

2010

Season of Publication

Summer

Paper Type

Master's Thesis

College

College of Arts and Sciences

Degree Name

Master of Science in Mathematical Sciences (MS)

Department

Mathematics & Statistics

First Advisor

Dr. Damon Hay

Second Advisor

Dr. Richard F. Patterson

Third Advisor

Dr. Daniel L. Dreibelbis

Department Chair

Dr. Scott Hochwald

College Dean

Dr. Barbara A. Hetrick

Abstract

This thesis starts with the fundamentals of matrix theory and ends with applications of the matrix singular value decomposition (SVD). The background matrix theory coverage includes unitary and Hermitian matrices, and matrix norms and how they relate to matrix SVD. The matrix condition number is discussed in relationship to the solution of linear equations. Some inequalities based on the trace of a matrix, polar matrix decomposition, unitaries and partial isometies are discussed. Among the SVD applications discussed are the method of least squares and image compression. Expansion of a matrix as a linear combination of rank one partial isometries is applied to image compression by using reduced rank matrix approximations to represent greyscale images. MATLAB results for approximations of JPEG and .bmp images are presented. The results indicate that images can be represented with reasonable resolution using low rank matrix SVD approximations.

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