The trace Cayley-Hamilton theorem

2025/10/23 by Grinberg, Darij
#15A15 #15A24 #FOS: Mathematics #History and Overview (math.HO) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2510.20689

Abstract

In this expository paper, various properties of matrix traces, determinants and adjugate matrices are proved, including the *trace Cayley-Hamilton theorem*, which says that kc_k + ∑_i=1^k Tr (Aⁱ) c_k-i = 0 for every k∈ℕ whenever A is an n× n-matrix with characteristic polynomial det (tI_n - A) = ∑_i=0ⁿ c_n-i tⁱ over a commutative ring \mathbbK. While the results are not new, some of the proofs are. The proofs illustrate some general techniques in linear algebra over commutative rings.

Citations

• Notes on the combinatorial fundamentals of algebra
• A combinatorial approach to matrix algebra

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