W&M ScholarWorks Undergraduate Honors Theses Theses, Dissertations, & Master Projects 6-2013 On Almost Normal Matrices Tyler J. Moran College of William and Mary Follow this and additional works at: https://scholarworks.wm.edu/honorstheses Part of the Mathematics Commons Recommended Citation Moran, Tyler J., "On Almost Normal Matrices" (2013). Undergraduate Honors Theses. Paper 574. https://scholarworks.wm.edu/honorstheses/574 This Honors Thesis is brought to you for free and open access by the Theses, Dissertations, & Master Projects at W&M ScholarWorks. It has been accepted for inclusion in Undergraduate Honors Theses by an authorized administrator of W&M ScholarWorks. For more information, please contact
[email protected]. On Almost Normal Matrices A thesis submitted in partial fulllment of the requirement for the degree of Bachelor of Science in Mathematics from The College of William and Mary by Tyler J. Moran Accepted for Ilya Spitkovsky, Director Charles Johnson Ryan Vinroot Donald Campbell Williamsburg, VA April 3, 2013 Abstract An n-by-n matrix A is called almost normal if the maximal cardinality of a set of orthogonal eigenvectors is at least n−1. We give several basic properties of almost normal matrices, in addition to studying their numerical ranges and Aluthge transforms. First, a criterion for these matrices to be unitarily irreducible is established, in addition to a criterion for A∗ to be almost normal and a formula for the rank of the self commutator of A. We then show that unitarily irreducible almost normal matrices cannot have flat portions on the boundary of their numerical ranges and that the Aluthge transform of A is never normal when n > 2 and A is unitarily irreducible and invertible.