Decomposition of matrices into products of commutators of quadratic matrices

Abstract

Let F be a field, p ( x ) be a quadratic polynomial in F [ x ] with a non-zero constant term, and n ≥ 2 be a positive integer. We say that T ∈ M n ( F ) is p ( x ) -quadratic if p ( T ) = 0 . In this paper, given A ∈ SL n ( F ) , we show that A can be decomposed into a product of at most two commutators of p ( x ) -quadratic matrices if either p ( x ) has two distinct roots in F and A is non-scalar, or if F is algebraically closed and has characteristic different from 2. A similar result for matrices over real quaternion division rings is also provided.