Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Common Invariant Subspace and Commuting Matrices   Let $K$ be a perfect field, $L$ be an extension field of $K$ and
$A,B\in\mathcal{M}_n(K)$. If $A$ has $n$ distinct eigenvalues in $L$ that are
explicitly known, then we can check if $A,B$ are simultaneously
triangularizable over $L$.
  Now we assume that $A,B$ have a common invariant proper vector subspace of
dimension $k$ over an extension field of $K$ and that $\chi_A$, the
characteristic polynomial of $A$, is irreducible over $K$. Let $G$ be the
Galois group of $\chi_A$. We show the following results i) If $k\in{1,n-1}$,
then $A,B$ commute. ii) If $1\leq k\leq n-1$ and $G=\mathcal{S}_n$ or
$G=\mathcal{A}_n$, then $AB=BA$. iii) If $1\leq k\leq n-1$ and $n$ is a prime
number, then $AB=BA$. Yet, when $n=4,k=2$, we show that $A,B$ do not
necessarily commute if $G$ is not $\mathcal{S}_4$ or $\mathcal{A}_4$. Finally
we apply the previous results to solving a matrix equation.

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The categories of this paper are: gr-qc
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The categories of this paper are: math-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: stat
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The categories of this paper are: nucl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: adap-org
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The categories of this paper are: mtrl-th
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The categories of this paper are: funct-an
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-ex
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The categories of this paper are: econ
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The categories of this paper are: nlin
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The categories of this paper are: q-alg
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The categories of this paper are: plasm-ph