On sums and products of diagonalizable matrices over division rings

TL;DR

This paper extends the study of diagonalizable matrices to noncommutative division rings, showing that such matrices can be expressed as sums or products of two diagonalizable matrices, and explores related Waring-type results.

Contribution

It generalizes previous results from fields to division rings, demonstrating new decompositions and establishing Waring-type theorems for matrices over noncommutative division rings.

Findings

Every matrix over a division ring can be written as a sum or product of two diagonalizable matrices.

The number 2 is not always valid for such decompositions under certain conditions.

New Waring-type results for matrices over division rings are established.

Abstract

This paper aims to continue the studies initiated by Botha in [Linear Algebra Appl. 273 (1998), 65-82; Linear Algebra Appl. 286 (1999), 37-44; Linear Algebra Appl. 315 (2000), 1-23] by extending them to matrices over noncommutative division rings. In particular, we show that every such matrix can be written as either a sum or a product of two diagonalizable matrices. The number $2$ is not valid under mild conditions on the center, similar to those in Botha's work on fields. By applying this result and other results obtained so far, we latter establish some Waring-type results for matrices.