Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8897957 | Linear Algebra and its Applications | 2018 | 17 Pages |
Abstract
Let Mn(F) be the algebra of nÃn 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 2nâ2. 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 2nâ2 is achieved.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Alexander Guterman, Thomas Laffey, Olga Markova, Helena Å migoc,