W&M ScholarWorks Undergraduate Honors Theses Theses, Dissertations, & Master Projects 5-2018 TP Matrices and TP Completability Duo Wang Follow this and additional works at: https://scholarworks.wm.edu/honorstheses Part of the Algebra Commons Recommended Citation Wang, Duo, "TP Matrices and TP Completability" (2018). Undergraduate Honors Theses. Paper 1159. https://scholarworks.wm.edu/honorstheses/1159 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]. TP Matrices and TP Completability A thesis submitted in partial fulfillment of the requirement for the degree of Bachelor of Science in Mathematics from The College of William and Mary by Duo Wang Committe Members: Charles Johnson Junping Shi Mark Greer Williamsburg, VA April 10, 2018 Abstract A matrix is called totally nonnegative (TN) if the determinant of every square submatrix is nonnegative and totally positive (TP) if the determinant of every square submatrix is positive. The TP (TN) completion problem asks which partial matrices have a TP (TN) completion. In this paper, several new TP-completable pat- terns in 3-by-n matrices are identified. The relationship between expansion and completability is developed based on the prior re- sults about single unspecified entry. These results extend our un- derstanding of TP-completable patterns. A new Ratio Theorem related to TP-completability is introduced in this paper, and it can possibly be a helpful tool in TP-completion problems.