#### Degree Name

Master of Science

#### Department or Program

Mathematics

#### Advisor

Fitzgerald, R.

#### Abstract

HAMDAN ALSULAIMANI, for the Master of Science in Mathematics, presented on NOV 6 2012, at Southern Illinois University Carbondale. TITLE: Diagonal (Triangular) Matrices PROFESSOR: Dr. R. Fitzgerald I present the Triangularization Lemma which says that let P be a set of properties, each of which is inherited by quotients. If every collection of transformations on a space of dimension greater than 1 that satisfies P is reducible, then every collection of transforma- tions satisfying P is triangularizable. I also present Burnside’s Theorem which says that the only irreducible algebra of linear transformations on the finite-dimensional vector space V of dimension greater than 1 is the algebra of all linear transformations mapping V into V. Moreover, I introduce McCoy’s Theorem which says that the pair {A,B} is triangularizable if and only if p(A,B)(AB-BA) is nilpotent for every noncommutative polynomial p. And then I show the relation between McCoy’s Theorem and Lie algebras.