Similarity of sets of matrices over a skew field

Abstract

This thesis looks at various questions in matrix theory over skew, fields. The common thread in all these considerations is the determination of an easily described form for a set of matrices, as simultaneously upper triangular or diagonal, for example.

The first chapter, in addition to giving some results which prove useful in later chapters, describes the work of P. M. Cohn on the normal form of a single matrix over a skew field. We use these results to show that, if the skew field D has a perfect center, then any matrix over D is similar to a matrix with entries in a commutative field.

The second chapter gives some results concerning commutativity, including the upper triangularizability of any set of commuting matrices, conditions allowing the simultaneous diagonalization of a set of commuting diagonalizable matrices, and a description, over skew fields with perfect centers, of matrices commuting with a given matrix. We end the chapter with a consideration of the problem of when a set of matrices over a skew field D is similar to a set of matrices with entries in a commutative sub-field of D.

The questions of simultaneously upper triangular-icing and diagonalizing semigroups of matrices are considered in the third chapter. A closure operation is defined on semigroups of matrices over a skew field, and it is shown that a semigroup is upper triangularizable (diagonalizable) if and only if its closure is. Necessary and sufficient conditions are then given for closed semigroups to the upper triangularizahle (diagonalizable).

The last chapter gives a few assorted results on groups of matrices, including the simultaneous upper triangularizability of a solvable group of unipotent matrices and a determination, for any skew field D, of those finite groups all of whose representations over D are diagonalizable.