On sums and products of diagonalizable matrices over division rings

2025/05/15 by Tran Nam Son, Son, Tran Nam
Computer Science · Mathematics · #Advanced Topics in Algebra #FOS: Mathematics #Matrix Theory and Algorithms #Rings and Algebras (math.RA) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2505.09909

openalex publication_date 2025/05/15 · openalex created_date 2025/10/15 · openalex updated_date 2026/07/28

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.

Citations

• Surjectivity of polynomial maps on Matrices
• Matrix Waring Problem -- II
• Matrix Waring Problem
• The Waring Problem for Matrix Algebras
• Sums and products of square-zero matrices
• What is the probability that a random integral quadratic form in n variables has an integral zero?
• Multiplicative groups of division rings

Related

• Matrices over commutative rings as sum of higher powers
• Products of Traceless and Semi-Traceless Matrices over Division Rings and their Applications
• The Waring Problem for Matrix Algebras
• Waring's problem for matrices over orders in algebraic number fields
• Trace and discriminant criteria for a matrix to be a sum of sixth and eighth powers of matrices