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I am aware of the theory of eigenvalues for matrices over fields. I was wondering to what extent this theory extends? Do we have a corresponding theory for matrices over integral domains, or at least over UFDs? H.C.Lee remarks here that there is no eigenvalue theory over general rings.

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    I know some of this can be done over commutative rings. The problem with anything noncommutative is that polynomials can have more roots than their degree. This means that the number of possible eigenvalues is not bounded by the dimensions of the matrix. It follows that most of spectral theory would not work.2011-12-05
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    @pki: Thanks. Do you have a reference for the theory over commutative rings?2011-12-05
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    Everything is exactly the same over integral domains (just pass to the field of fractions).2011-12-07
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    @Qiaochu Yuan: How does one make sure that the eigenvalues lie in the ring and not in the fraction field?2011-12-07
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    @B M: one can't. This isn't even true over fields; in general you need to pass to the algebraic closure.2011-12-07
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    If I recall, polynomials over the quaternions have "roots" that are either points or spheres. So it is a nice generalization of complex polynomials, just a fair bit more general.2011-12-07
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    For a matrix $M\in M_n(A)$ where $A$ is a domain, the characteristic polynomial $P(M)$ has coefficients in $A$ and is monic. Thus, the roots of $P(M)$ all lie in an integral closure of $A$. Therefore, we can be sure that the roots belong to $A$ only if $A$ is integrally closed.2011-12-12

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One generalization of the notion of eigenvalue can be found in module theory. One thinks of a linear operator $T : V \to V$ acting on a vector space $V$ as a module over the polynomial ring $k[T]$, and then an eigenvector $Tv = \lambda v$ spans a simple submodule of $V$. More generally, the generalized eigenspace associated to an eigenvalue $\lambda$ is an indecomposable submodule of $V$. The statement that $T$ has a Jordan normal form is then subsumed under the general theory of finitely-generated modules over principal ideal domains.

Generalizing, one may think of an $n \times n$ matrix over an arbitrary ring $R$ acting on column vectors over $R$ as describing an endomorphism of $R^n$ as a right $R$-module. This gives $R^n$ the additional structure of a left $R[T]$-module, and one can apply the general tools of module theory to study this module. Of course, if $R$ is complicated then the corresponding theory will be complicated.