## Abstract

Let M_{n}(F) be the algebra of n×n matrices over the field F and let S be a generating set of M_{n}(F) as an F-algebra. The length of a finite generating set S of M_{n}(F) is the smallest number k such that words of length not greater than k generate M_{n}(F) as a vector space. It is a long standing conjecture of Paz that the length of any generating set of M_{n}(F) cannot exceed 2n−2. We prove this conjecture under the assumption that the generating set S contains a nonderogatory matrix. In addition, we find linear bounds for the length of generating sets that include a matrix with some conditions on its Jordan canonical form. Finally, we investigate cases when the length 2n−2 is achieved.

Original language | English |
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Pages (from-to) | 234-250 |

Number of pages | 17 |

Journal | Linear Algebra and Its Applications |

Volume | 543 |

DOIs | |

State | Published - 15 Apr 2018 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2018 Elsevier Inc.

## Keywords

- Finite-dimensional algebras
- Lengths of sets and algebras
- Nonderogatory matrices
- Paz's conjecture