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Brown's Matrices over Commutative Rings book discusses the theory of eigenvalues, eigenvectors, and diagonalizing matrices over commutative rings, but unless I've missed something, nothing like generalized eigenvectors is even mentioned. Has the notion of generalized eigenvectors ever been studied over commutative rings in any context more general than working over a field?

My interest stems from trying to find a closed form expression for the matrix exponential $e^{zA}$, where $z$ is an element of a finite dimensional real associative commutative algebra, and $A$ is a matrix built of elements of said algebra, which in the real case relies on the theory of generalized eigenvectors.

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Your question is how to compute the matrix exponential $\exp(a)$ where $a$ is an element of a finite-dimensional real associative algebra $A$. (Your phrasing is not more general: you're implicitly working in the algebra $M_n(A)$.) This reduces to the case of real matrices, because the left regular representation

$$A \ni a \mapsto \left( L_a : b \mapsto ab \right) \in \text{End}_{\mathbb{R}}(A) \cong M_{\dim A}(\mathbb{R})$$

is always faithful, and

$$L_{\exp(a)} = \exp(L_a).$$

So there's no need for a more general theory.

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