Date of Award

Spring 5-2013

Degree Type

Masters Thesis

Degree Name

Master of Science (MS)

Department

Mathematics

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.

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