An improved bound for the lengths of matrix algebras

Abstract

Let S be a set of n × n matrices over a field F. We show that the F-linear span of the words in S of length at most 2n log 2 n + 4n is the full F-algebra generated by S. This improves on the n 2 /3 + 2/3 bound by Paz (1984) and an O(n 3/2 ) bound of Pappacena (1997).Let S be a subset of a finite-dimensional associative algebra A over a field F. An element a ∈ A is said to be a word of length k in S if there are a 1 , . . . , a k ∈ S such that a = a 1 . . . a k . We denote the set of all such words by S k , and we write FS k for the F-linear span of S k . Similarly, FS k will stand for the F-linear span of all the words in S that have length at most k.Definition 1. The length ℓ(S) is the smallest integer k for which FS k is the full subalgebra generated by S. We also define ℓ(A) as the maximum value of ℓ(S), where S runs over all subsets of A that generate A as an F-algebra.

An elementary method to compute the algebra generated by some given matrices and its dimension

“…It is clear that the choice of k = n 2 works. If we believe the Paz conjecture [9], one could take k = 2n − 2, although the best known bounds, obtained very recently by Shitov in [10], give that we can choose k = 2n log 2 n + 4n, improving the best previously known bounds by Pappacena [8]. All such methods necessitate a slowdown required to evalute such an exponentiation.…”

Quantum Version of Wielandt’s Inequality Revisited

“…Throughout this section we work under the assumption that dim L j = D 2 for j large enough. We start with a general lemma taken from [Shi18].…”

“…He was able to prove an upper bound of D 2 /3+2/3, which was later improved to O D 1.5 by Pappacena [Pap97]. The latest best known bound is O(D log D) by the second author [Shi18]. The 2D − 2 conjecture is known to hold if L contains a non-derogatory matrix [GLMŠ18] and for D 5 (see [Shi18]).…”

“…The latest best known bound is O(D log D) by the second author [Shi18]. The 2D − 2 conjecture is known to hold if L contains a non-derogatory matrix [GLMŠ18] and for D 5 (see [Shi18]). This approach leads to our main theorem:…”

Local theory of free noncommutative functions: germs, meromorphic functions, and Hermite interpolation

“…, T g . By[Shi+, Theorem 3], A is generated by polynomials in T of degree 2N log 2 N + 4N − 4. Therefore there exists f ∈ <x> of degree at most2N log 2 N + 4N − 4 such that f ℓ = ∆ ℓ Y f = ∆ ℓ Y f for ℓ ≤ L.Remark6.14.…”