Matrix Similarity over Commutative Rings and Remarks on the Gauger-Byrnes Theorem
DOI:
https://doi.org/10.29020/nybg.ejpam.v19i2.7561Keywords:
Matrix similarity, invariant factors, Kronecker product, minimal polynomial, characteristic polynomial.Abstract
We develop a unified, module-theoretic approach to matrix similarity over a unital commutative ring R. For A, B ∈ Rn, we show that A and B are similar over R if and only if Ker(τAB), with τAB(Z) = ZA − BZ, is a right free C(A)-module of rank 1 whose generators are invertible in Rn; equivalently, the same condition holds for the left C(B)-module structure. This viewpoint provides a complete and intrinsic characterization of similarity over arbitrary commutative rings, leads to a concise proof of an enhanced Gauger–Byrnes theorem over fields, and clarifies the role of Kronecker operators and centralizer modules, while bypassing invariant-factor computations.
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