## An improved bound for the lengths of matrix algebras.(English)Zbl 1419.15018

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)].

### 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
Full Text:

### 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
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