Expressing upper triangular matrices as products of commutators of finite order elements

Expressing Matrices Into Products of Commutators of Involutions, Skew-Involutions, Finite Order and Skew Finite Order Matrices

“…Here we can observe that B 2k = −I and C 2k = I. From the Lemma 3.6 in Gargate [2] we can observe that, if B 2k = −I in the proof of this Lemma, we obtain that (BC) 2k = −F 2k (B, C) where F 2k (B, C) is the product of 4k − 3 commutators whose entries are powers of B and C. In this case we define A = −(BC) 2k and obtain the result. Lemma 3.5.…”

“…We can observe that B k = C k = I and similarly to Lemma 3,5 in Gargate [2] we define A = (BC) k that is product of 2k − 3 commutators. 4.)…”

“…Proof of Theorem 1.3. In this case, we can rewrite A = B k = (CD) k with C k = D k = I and the result follows from Lemma 3.6 in Gargate [2].…”

“…Considering the results obtained in Gargate [2], we have the following Theorem 1.5. All element in SL n (C) can be written as a product of at most…”

“…They prove that every unitriangular matrix of a specified form is a commutator of two other unitriangular matrices. Considering ω a kth root of unity in R, recently Gargate in [2] prove that every element of the group U T ∞ (R) can be expressed as a product of 4k − 6 commutators all depending of powers of elements in U T (k) ∞ (R) of order k. In the section 3 following the same direction, we study the subgroup U T n (m, R) and U T ∞ (m, R) of U T n (R) and U T ∞ (m, R) respectively which have zero entries on the first m − 1 superdiagonals.…”