#### Date of Award

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 *A*^{2} = *B*^{2}, *AB* = *A*^{2} and *AB* = *B*^{2}. 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 *A*^{2} = *B*^{2} (when *A* has many distinct eigenvalues λ_{1}, λ_{2}, ... , λ_{k} such that λ_{i} ≠ -λ_{j}) and in the case of *AB* = *A*^{2}, 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 *A*^{2} = *B*^{2}.

#### Copyright

2013, Thir Raj Dangal

#### Recommended Citation

Dangal, Thir Raj, "Solutions of Matrix Equations" (2013). *Master's Theses*. 455.

https://aquila.usm.edu/masters_theses/455

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