Products of Traceless and Semi-Traceless Matrices over Division Rings and their Applications

TL;DR

This paper investigates the decomposition of matrices over division rings into products of traceless or semi-traceless matrices, establishing bounds and exploring applications in algebraic decompositions.

Contribution

It provides new bounds on the number of traceless and semi-traceless matrices needed to represent any matrix over a division ring, and applies these results to algebraic decompositions.

Findings

Every matrix over a division ring is a product of at most twelve traceless matrices.

Every matrix over a division ring is a product of at most four semi-traceless matrices.

Elements of certain finite-dimensional algebras can be decomposed into products of at most four generalized commutators.

Abstract

We study the problem when every matrix over a division ring is representable as either the product of traceless matrices or the product of semi-traceless matrices, and also give some applications of such decompositions. Specifically, we establish the curious facts that every matrix over a division ring is a product of at most twelve traceless matrices as well as a product of at most four semi-traceless matrices. We also examine finitary matrices and certain images of non-commutative polynomials by applying the obtained so far results showing that the elements of some finite-dimensional algebras over a special field as well as that these of the matrix algebra over any division ring possess some rather interesting and non-trivial decompositions into products of at most four generalized commutators.