Matrix similarity

Abstract: This thesis will deal with similar matrices, also referred to as matrix conju- gation. The rst problem we will attack is whether or not two given matrices are similar over some eld. To solve this problem we will introduce the Ratio- nal Canonical Form, RCF. From this normal form, also called the Frobenius normal form, we can determine whether or not the given matrices are sim- ilar over any eld. We can also, given some eld F, see whether they are similar over F or not. To be able to understand and prove the existence and uniqueness of the RCF we will introduce some additional module theory. The theory in this part will build up to nally prove the theorems regarding the RCF that can be used to solve our problem. The next problem we will investigate is regarding simultaneous conjugation, i.e. conjugation by the same matrix on a pair of matrices. When are two pairs of similar matrices simultaneously conjugated? Can we nd any necessary or even sucient conditions on the matrices? We will address this more complicated issue with the theory assembled in the rst part. 2