On the length of generating sets with conditions on minimal polynomial

Linear upper bounds may be derived by imposing specific structural conditions on a generating set, such as additional constraints on ranks, eigenvalues, or the degree of the minimal polynomial of the generating matrices. This paper establishes a linear upper bound of $3n-5$ for generating sets that contain a matrix whose minimal polynomial has a degree exceeding $\frac{n}{2}$, where $n$ denotes the order of the matrix. Compared to the bound provided in \cite[Theorem 3.1]{r2}, this result reduces the constraints on the Jordan canonical forms. Additionally, it is demonstrated that the bound $\frac{7n}{2}-4$ holds when the generating set contains a matrix with a minimal polynomial of degree $t$ satisfying $2t\le n\le 3t-1$. The primary enhancements consist of quantitative bounds and reduced reliance on Jordan form structural constraints.