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On sums and products of diagonalizable matrices over division rings

Tran Nam Son

TL;DR

The paper generalizes the matrix decomposition problem of expressing elements as sums or products of diagonalizable matrices from fields to noncommutative division rings. It develops a framework based on companion matrices, Sylvester equations, and rational canonical forms to prove that every matrix can be written as a sum or product of two diagonalizable matrices under suitable center-size conditions, with tight sharpness results (2 or 3) depending on the center. It then derives Waring-type results for matrices over radicable and algebraically closed division rings, showing surjectivity of diagonal-polynomial maps and expressing arbitrary matrices as sums, products, or linear combinations of few powers of matrices. These results illuminate how the center and algebraic structure of the division ring govern decompositions, and connect matrix decompositions to broader algebraic themes such as group algebras and rational canonical forms in the noncommutative setting.

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.

On sums and products of diagonalizable matrices over division rings

TL;DR

The paper generalizes the matrix decomposition problem of expressing elements as sums or products of diagonalizable matrices from fields to noncommutative division rings. It develops a framework based on companion matrices, Sylvester equations, and rational canonical forms to prove that every matrix can be written as a sum or product of two diagonalizable matrices under suitable center-size conditions, with tight sharpness results (2 or 3) depending on the center. It then derives Waring-type results for matrices over radicable and algebraically closed division rings, showing surjectivity of diagonal-polynomial maps and expressing arbitrary matrices as sums, products, or linear combinations of few powers of matrices. These results illuminate how the center and algebraic structure of the division ring govern decompositions, and connect matrix decompositions to broader algebraic themes such as group algebras and rational canonical forms in the noncommutative setting.

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 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.
Paper Structure (11 sections, 21 theorems, 79 equations)

This paper contains 11 sections, 21 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Sums of diagonalizable matrices
  3. Division rings with center containing at least three elements
  4. Division rings with center containing only two elements
  5. Products of diagonalizable matrices
  6. Division rings of characteristic different from 2
  7. Division rings of characteristic 2
  8. Division rings with center containing sufficiently many elements
  9. Some Waring-type results for images of polynomials
  10. Division rings with radicable multiplicative groups
  11. Algebraically closed division rings

Key Result

Lemma 2.2

Bo_Bo_22 Let $A \in \mathrm{M}_n(D)$ and $B \in \mathrm{M}_m(D)$, where $n$ and $m$ are positive integers. Suppose there exists a polynomial $p$ in one variable with coefficients in the center of $D$ such that $p(A) = 0$ and $p(B)$ is invertible. Then, for any $C \in \mathrm{M}_{n \times m}(D)$, the has a unique solution $X \in \mathrm{M}_{n \times m}(D)$.

Theorems & Definitions (28)

  • Remark 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • proof
  • Theorem 2.8
  • Theorem 2.9
  • ...and 18 more