Two n x n matrices A and B can make many different matrix equations, e.g., AB = BA, AB = AA, ABA = BAB, and AAB = BAA. It is not always easy to describe solutions to these matrix equations. This thesis considers the problem of describing solutions to the matrix equations A2 = B2, AB = A2 and AB = B2. This problem is motivated by considering the properties of commutative matrices (i.e., AB = BA), as well as the matrix form of the Yang-Baxter equation, ABA = BAB.
For each of these equations, solutions are provided such that A ≠ B. However, in the case of A2 = B2 (when A has many distinct eigenvalues λ1, λ2, ... , λk such that λi ≠ -λj) and in the case of AB = A2, the matrices A and B must have a common eigenvector. In addition, matrices arising from graphs are considered, and restrictions are determined which impIy unique solutions to the matrix equation A2 = B2.
","abstract_html":"<p>Two <em>n</em> x <em>n</em> matrices A and B can make many different matrix equations, e.g., <em>AB</em> = <em>BA</em>, <em>AB</em> = <em>AA</em>, <em>ABA</em> = <em>BAB</em>, and <em>AAB</em> = <em>BAA</em>. It is not always easy to describe solutions to these matrix equations. This thesis considers the problem of describing solutions to the matrix equations <em>A</em><sup>2</sup> = <em>B</em><sup>2</sup>, <em>AB</em> = <em>A</em><sup>2</sup> and <em>AB</em> = <em>B</em><sup>2</sup>. This problem is motivated by considering the properties of commutative matrices (i.e., <em>AB</em> = <em>BA</em>), as well as the matrix form of the Yang-Baxter equation, <em>ABA</em> = <em>BAB</em>.</p> <p>For each of these equations, solutions are provided such that A ≠ B. However, in the case of <em>A</em><sup>2</sup> = <em>B</em><sup>2</sup> (when <em>A</em> has many distinct eigenvalues λ<sub>1</sub>, λ<sub>2</sub>, ... , λ<sub>k</sub> such that λ<sub>i</sub> ≠ -λ<sub>j</sub>) and in the case of <em>AB</em> = <em>A</em><sup>2</sup>, the matrices <em>A</em> and <em>B</em> must have a common eigenvector. In addition, matrices arising from graphs are considered, and restrictions are determined which impIy unique solutions to the matrix equation <em>A</em><sup>2</sup> = <em>B</em><sup>2</sup>. </p>","abstract_has_math":false,"creators":["Dangal, Thir Raj"],"institution":null,"degree_name":"Master of Science (MS)","degree_level":"Masters Thesis","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013-05-01T07:00:00Z","date_published":"2013-05-01T07:00:00Z","updated_at":"2026-07-24T05:44:58Z","subjects":[],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://aquila.usm.edu/masters_theses/455","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Dangal, Thir Raj"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2018-11-05T08:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Masters Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science (MS)"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://aquila.usm.edu/masters_theses/455"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Two n x n matrices A and B can make many different matrix equations, e.g., AB = BA, AB = AA, ABA = BAB, and AAB = BAA. It is not always easy to describe solutions to these matrix equations. This thesis considers the problem of describing solutions to the matrix equations A2 = B2, AB = A2 and AB = B2. This problem is motivated by considering the properties of commutative matrices (i.e., AB = BA), as well as the matrix form of the Yang-Baxter equation, ABA = BAB.
For each of these equations, solutions are provided such that A ≠ B. However, in the case of A2 = B2 (when A has many distinct eigenvalues λ1, λ2, ... , λk such that λi ≠ -λj) and in the case of AB = A2, the matrices A and B must have a common eigenvector. In addition, matrices arising from graphs are considered, and restrictions are determined which impIy unique solutions to the matrix equation A2 = B2.
"]},{"key":"dc:title","label":"Title","values":["Solutions of Matrix Equations"]}]}],"canonical_facts":{"dc:creator":["Dangal, Thir Raj"],"dc:date.available":["2018-11-05T08:00:00Z"],"dc:description.abstract":["Two n x n matrices A and B can make many different matrix equations, e.g., AB = BA, AB = AA, ABA = BAB, and AAB = BAA. It is not always easy to describe solutions to these matrix equations. This thesis considers the problem of describing solutions to the matrix equations A2 = B2, AB = A2 and AB = B2. This problem is motivated by considering the properties of commutative matrices (i.e., AB = BA), as well as the matrix form of the Yang-Baxter equation, ABA = BAB.
For each of these equations, solutions are provided such that A ≠ B. However, in the case of A2 = B2 (when A has many distinct eigenvalues λ1, λ2, ... , λk such that λi ≠ -λj) and in the case of AB = A2, the matrices A and B must have a common eigenvector. In addition, matrices arising from graphs are considered, and restrictions are determined which impIy unique solutions to the matrix equation A2 = B2.
"],"dc:identifier":["https://aquila.usm.edu/masters_theses/455"],"dc:title":["Solutions of Matrix Equations"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Masters Thesis"],"thesis:degree_name":["Master of Science (MS)"]},"updated_at":"2026-07-24T05:44:58Z"}