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For the pairs of quasi-commuting matrices we completely characterize natural numbers that can be realized as the lengths of these pairs of generators.

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 2 n − 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 2 n − 2 is achieved.

Let M n ( F ) be the algebra of n × n matrices 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. Traditionally the identity matrix is assumed to be automatically included in all generating sets S and counted as a word of length 0. In this paper we discuss how the problem changes if this assumption is removed.

In this paper we discuss some partial solutions of the length conjecture which describes the length of a generating system for matrix algebras. We consider mainly the sets of two matrices which are quasi-commuting. It is shown that in this case the length function is linearly bounded. We also analyze which particular natural numbers can be realized as the lengths of certain special generating sets and prove that for commuting or product-nilpotent pairs all possible numbers are realizable, however there are non-realizable values between lower and upper bounds for the other quasi-commuting pairs. In conclusion we also present several related open problems.