Open AccessDecomposition of matrices into commutators of unipotent matrices of index 2
Author(s) -
Xin Hou
Publication year - 2021
Publication title -
the electronic journal of linear algebra
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.447
H-Index - 31
eISSN - 1537-9582
pISSN - 1081-3810
DOI - 10.13001/ela.2021.5641
Subject(s) - unipotent , mathematics , complex matrix , matrix (chemical analysis) , field (mathematics) , decomposition , product (mathematics) , pure mathematics , combinatorics , group (periodic table) , geometry , physics , chemistry , organic chemistry , chromatography , quantum mechanics
Let $\mathbb{C}$ be the complex field. Denote by $\mathrm{SL}_n(\mathbb{C})$ the group of all complex $n\times n$ matrices with determinant $1$. It is proved that every matrix in $\mathrm{SL}_n(\mathbb{C})$ can be decomposed into a product of two commutators of unipotent matrices of index $2$. Moreover, two is the smallest such number.