On the Length of Generating Sets with Conditions on Minimal Polynomial

Abstract

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.

Keywords

The full matrix algebra / Length of an algebra / Generating systems / The degree of minimal polynomial / 15A03 / 15A30 / 16P10

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