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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.17348 |
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Table of Contents:
- 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.