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

The Slot Length of a Family of Matrices

“…A recent result [9,Theorem 3] of Shitov shows that the length is at most 2n log 2 n + 4n − 4. For other recent results and further references, see [1,7].…”

“…Let A be a finite family of distinct symbols. Every word w in A can be written, using index notation, uniquely in the form a r 1 1 a r 2 2 • • • a r k−1 k−1 a r k k , where each r i is a positive integer and adjacent bases are distinct elements of A, that is, a i a i+1 for 1 ≤ i ≤ k − 1. We define the slot length of the word w to be k. For example, if a, b, c, x, y, z ∈ A, a 3 has slot length 1; a 2 bc 8 has slot length 3, assuming that a b c; x 5 y 4 z 4 a 2 bc 5 has slot length 6, assuming that x y z a b c.…”

-Minimum Spanning Lengths and an Extension to Burnside’s Theorem on Irreducibility

“…Let W be a nonempty word in S, written in its standard form, using index notation wherever possible. Thus, W = x s 1 1 x s 2 2 . .…”

“…Let A be a finite nonempty set of distinct n × n matrices over a field F. For every k ∈ Z + , let 1 V k (A), or simply 1 V k , be the linear span of the 1-words in A of length at most k. Then 1 V k ⊆ 1 V k+1 for every k ≥ 1. There exists a smallest positive integer K 1 such that 1…”

“…Still with A and F as above, the (earlier) notion of 'length of A' has been considered by several authors, especially since Paz's paper [11] in 1984 (see [1,3,4,7,9,10,12]). For every integer k ≥ 0, we let V k (A) denote the linear span of the words in A of length at most k, taking the unique word of length zero, called the empty word, to be the identity matrix.…”