Products of commutators of unipotent matrices of index $2$ in $\mathrm{GL}_n(\mathbb H)$

Yıl 2024, , 121 - 133, 12.07.2024

Ha Nguyen Thi Thai , Dao Trong Toan

https://doi.org/10.24330/ieja.1476670

Öz

The aim of this paper is to show that if $\mathbb{H}$ is the real quaternion division ring and $n$ is an integer greater than $1,$ then every matrix in the special linear group $\mathrm{SL}_n(\mathbb{H})$ can be expressed as a product of at most three commutators of unipotent matrices of index $2$.

Anahtar Kelimeler

Matrix decomposition, matrix over a division ring, commutator length

Kaynakça

• M. H. Bien, T. H. Dung and N. T. T. Ha, A certain decomposition of infinite invertible matrices over division algebras, Linear Multilinear Algebra, 71 (2023), 1948-1956.
• M. H. Bien, T. H. Dung, N. T. T. Ha and T. N. Son, Decompositions of matrices over division algebras into products of commutators, Linear Algebra Appl., 646 (2022), 119-131.
• M. H. Bien, T. H. Dung, N. T. T. Ha and T. N. Son, Involution widths of skew linear groups generated by involutions, Linear Algebra Appl., 679 (2023), 305-326.
• M. H. Bien, T. N. Son, P. T. T. Thuy and L. Q. Truong, Products of unipotent matrices of index 2 over division rings, Submitted.
• E. W. Ellers and J. Malzan, Products of reflections in GL(n;H), Linear Multilinear Algebra, 20(4) (1987), 281-324.
• N. T. T. Ha, P. H. Nam and T. N. Son, Products of commutators of involutions in skew linear groups, Acta Math. Vietnam., Submitted.
• X. Hou, Decomposition of infinite matrices into products of commutators of involutions, Linear Algebra Appl., 563 (2019), 231-239.
• X. Hou, Decomposition of matrices into commutators of unipotent matrices of index 2, Electron. J. Linear Algebra, 37 (2021), 31-34.
• X. Hou, S. Li and Q. Zheng, Expressing infinite matrices over rings as products of involutions, Linear Algebra Appl., 532 (2017), 257-265.
• J. P. E. Joven and A. T. Paras, Products of skew-involutions, Electron. J. Linear Algebra, 39 (2023), 136-150.
• A. Kanel-Belov, B. Kunyavskii and E. Plotkin, Word equations in simple groups and polynomial equations in simple algebras, Vestnik St. Petersburg Univ. Math., 46 (2013), 3-13.
• T. Y. Lam, A First Course in Noncommutative Rings, 2nd edition, Graduate Texts in Mathematics, 131, Springer-Verlag, New York, 2001.
• D. W. Lewis, Quaternion algebras and the algebraic legacy of Hamilton's quaternions, Irish Math. Soc. Bull., 57 (2006), 41-64.
• T. A. Loring, Factorization of matrices of quaternions, Expo. Math., 30 (2012), 250-267.
• D. I. Merino and V. V. Sergeichuk, Littlewood's algorithm and quaternion matrices, Linear Algebra Appl., 298 (1999), 193-208.
• I. Niven, Equations in quaternions, Amer. Math. Monthly, 48 (1941), 654-661.
• C. de Seguins Pazzis, The sum and the product of two quadratic matrices: regular cases, Adv. Appl. Clifford Algebr., 32 (2022), 54 (43 pp).
• T. N. Son, T. H. Dung, N. T. T. Ha and M. H. Bien, On decompositions of matrices into products of commutators of involutions, Electron. J. Linear Algebra, 38 (2022), 123-130.
• L. N. Vaserstein and E. Wheland, Commutators and companion matrices over rings of stable rank 1, Linear Algebra Appl., 142 (1990), 263-277.
• J. Voight, Quaternion Algebras, Graduate Texts in Mathematics, Vol. 288, Springer, 2021.
• Z. X. Wan, Geometry of Matrices, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
• G. Wang, T. Jiang, V. I. Vasil'ev and Z. Guo, On singular value decomposition for split quaternion matrices and applications in split quaternionic mechanics, J. Comput. Appl. Math., 436 (2024), 115447 (10 pp).
• J. H. Wang and P. Y. Wu, Products of unipotent matrices of index 2, Linear Algebra Appl., 149 (1991), 111-123.
• H. You, Decomposition of matrices into commutators of reflections, J. Syst. Sci. Complex., 16(1) (2003), 67-73.
• F. Zhang, Quaternions and matrices of quaternions, Linear Algebra Appl., 251 (1997), 21-57.
• B. Zheng, Decomposition of matrices into commutators of involutions, Linear Algebra Appl., 347 (2002), 1-7.