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2 results on '"Paz conjecture"'

1. Sweeping words and the length of a generic vector subspace of [formula omitted].

Author

Klep, Igor and Špenko, Špela

Subjects
*MULTIGRAPH, *MATRICES (Mathematics), *COMBINATORICS, *SUBSPACES (Mathematics), *INFINITY (Mathematics), *SET theory
Abstract

The main result of this short note is a generic version of Paz' conjecture on the length of generating sets in matrix algebras. Consider a generic g -tuple A _ = ( A 1 , … , A g ) of n × n matrices over an infinite field. We show that whenever g 2 d ≥ n 2 , the set of all words of degree 2 d in A _ spans the full n × n matrix algebra. Our proofs use generic matrices, are combinatorial and depend on the construction of special kinds of directed multigraphs with few edge-disjoint walks. [ABSTRACT FROM AUTHOR]

Published
2016
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2. Sweeping words and the length of a generic vector subspace of M_n(F)

Author

Špela Špenko, Igor Klep, Mathematics, and Algebra

Subjects
Local linear independence, Algèbre linéaire et matricielle, 010103 numerical & computational mathematics, Paz conjecture, Mathematical proof, 01 natural sciences, Theoretical Computer Science, Combinatorics, Set (abstract data type), Matrix (mathematics), Free algebra, Generic matrix, Discrete Mathematics and Combinatorics, 0101 mathematics, Directed multigraph, Mathematics, Discrete mathematics, Conjecture, Degree (graph theory), Length of a vector space, 010102 general mathematics, Mathematics - Rings and Algebras, Linear subspace, Discriminant, Computational Theory and Mathematics
Abstract

The main result of this short note is a generic version of Paz's conjecture on the lengths of generating sets in matrix algebras. Consider a generic g-tuple A=(A_1,..., A_g) of nxn matrices over a field. We show that whenever $g^{2d}\geq n^2$, the set of all words of degree 2d in A spans the full nxn matrix algebra. Our proofs use generic matrices, are combinatorial and depend on the construction of a special kind of directed multigraphs with few edge-disjoint walks., Comment: 8 pages

Published
2015

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