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 Theorem 3.1 of Guterman et al. (Linear Algebra Appl 543: 234–250, 2018), this result reduces the constraints on the Jordan canonical forms. In addition, 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 \(2t\leqslant n\leqslant 3t-1\) . The primary enhancements consist of quantitative bounds and reduced reliance on Jordan form structural constraints.
