Shitov, Yaroslav An improved bound for the lengths of matrix algebras. (English) Zbl 1419.15018 Algebra Number Theory 13, No. 6, 1501-1507 (2019). Summary: Let \(S\) be a set of \(n\times n\) matrices over a field \(\mathbb{F}\). We show that the \(\mathbb{F}\)-linear span of the words in \(S\) of length at most \[ 2n\log_2n+4n \] is the full \(\mathbb{F}\)-algebra generated by \(S\). This improves on the \(\frac{n^2}{3}+\frac{2}{3}\) bound by A. Paz [Linear Multilinear Algebra 15, 161–170 (1984; Zbl 0536.15007)] and an \(O(n^{3/2})\) bound of C. J. Pappacena [J. Algebra 197, No. 2, 535–545 (1997; Zbl 0888.16008)]. Cited in 13 Documents MSC: 15A54 Matrices over function rings in one or more variables 15A30 Algebraic systems of matrices 16P10 Finite rings and finite-dimensional associative algebras Keywords:matrix theory; finite-dimensional algebras; generating sets Citations:Zbl 0536.15007; Zbl 0888.16008 PDF BibTeX XML Cite \textit{Y. Shitov}, Algebra Number Theory 13, No. 6, 1501--1507 (2019; Zbl 1419.15018) Full Text: DOI arXiv OpenURL References: [1] 10.1016/j.laa.2018.01.002 · Zbl 1382.15027 [2] 10.1016/0024-3795(86)90311-3 · Zbl 0609.15004 [3] 10.1017/S0004972709000112 · Zbl 1184.15014 [4] 10.1017/S0004972700035462 · Zbl 1102.15012 [5] 10.4213/rm1653 [6] 10.1109/TIT.2019.2897772 · Zbl 1432.82007 [7] 10.1006/jabr.1997.7140 · Zbl 0888.16008 [8] 10.1017/S0017089503001204 · Zbl 1048.16011 [9] 10.1080/03081088408817585 · Zbl 0536.15007 [10] 10.1007/978-1-4612-1200-3 · Zbl 0981.15007 [11] 10.1109/TIT.2010.2054552 · Zbl 1366.81131 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.