This thesis would not have been possible without the help of my advisor, Dr. Robert Gilmore. I would like to thank him for suggesting such a fascinating topic for my thesis and for his patience in mentoring me and helping me to grasp the ideas. Thank you for encouraging me to keep reading, knowing that I would not understand anything in the beginning, and for taking such a keen interest in my thesis. I really enjoyed our conversations and looked forward to them each week. They were tremendously helpful for understanding the big picture.I would also like to thank Dr. Robert Boyer for allowing me attend his graduate course on Lie groups this spring. The lectures were helpful for understanding the mathematical background when writing Chapter 2.Finally, I would like to thank my family for their love and support throughout my time at Drexel.

An n × n matrix A has a normal defect of k if there exists an (n + k) × (n + k) normal matrix A ext with A as a leading principal submatrix and k minimal. In this paper we compute the normal defect of a special class of 4 × 4 matrices, namely matrices whose only nonzero entries lie on the superdiagonal, and we provide details for constructing minimal normal completion matrices A ext . We also prove a result for a related class of n × n matrices. Finally, we present an example of a 6 × 6 block diagonal matrix having the property that its normal defect is strictly less than the sum of the normal defects of each of its blocks, and we provide sufficient conditions for when the normal defect of a block diagonal matrix is equal to the sum of the normal defects of each of its blocks.