Decompositions of Matrices Into a Sum of Torsion Matrices and Matrices of Fixed Nilpotence

For n≥ 2 and fixed k≥ 1, we study when a square matrix A over an arbitrary field 𝔽 can be decomposed as T+N where T is a torsion matrix and N is a nilpotent matrix with N^k=0. For fields of prime characteristic, we show that this decomposition holds as soon as the characteristic polynomial of A∈𝕄_n(𝔽) is algebraic over its base field and the rank of A is at least n/k, and we present several examples that show that the decomposition does not hold in general. Furthermore, we completely solve this decomposition problem for k=2 and nilpotent matrices over arbitrary fields (even over division rings). This somewhat continues our recent publications in Lin. & Multilin. Algebra (2023) and Internat. J. Algebra & Computat. (2022) as well as it strengthens results due to Calugareanu-Lam in J. Algebra & Appl. (2016).