A resolution of Paz's conjecture in the presence of a nonderogatory matrix

Alexander Guterman, Thomas Laffey, Olga Markova, Helena Šmigoc

Research output: Contribution to journalArticlepeer-review

26 Scopus citations


Let Mn(F) be the algebra of n×n matrices over the field F and let S be a generating set of Mn(F) as an F-algebra. The length of a finite generating set S of Mn(F) is the smallest number k such that words of length not greater than k generate Mn(F) as a vector space. It is a long standing conjecture of Paz that the length of any generating set of Mn(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 languageEnglish
Pages (from-to)234-250
Number of pages17
JournalLinear Algebra and Its Applications
StatePublished - 15 Apr 2018
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2018 Elsevier Inc.


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


Dive into the research topics of 'A resolution of Paz's conjecture in the presence of a nonderogatory matrix'. Together they form a unique fingerprint.

Cite this