= Burnside matrix-algebra theorem
{c}
{title2=$A\subseteq\operatorname{End}_{\mathbb C}(V),\quad V\text{ irreducible}\ \Longrightarrow\ A=\operatorname{End}_{\mathbb C}(V)$}
For a nonzero finite-dimensional complex <vector space> $V$, a unital <matrix algebra> acting irreducibly on $V$ is the full endomorphism algebra. Its <Jacobson radical> annihilates $V$: otherwise the radical times $V$ is all of $V$, contradicting nilpotence. Faithfulness kills the radical, and the <Artin–Wedderburn theorem> then leaves one simple matrix factor acting on its standard column <module>. This is a matrix-algebra theorem, distinct from the orbit-counting <Burnside lemma> and the finite-group <Burnside's theorem>.
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