----- > [!proposition] Proposition. ([[representation of finite abelian group consists of simultaneously diagonalizable operators]]) > Let $G$ be a finite [[abelian group]] [[group action|acting on]] a finite-dimensional [[vector space]] $V$ via [[group representation|representation]] $(\rho, V)$. Then the [[linear operator|operators]] $\{ \rho_{g} \}_{g \in G}$ are [[simultaneously diagonalizable]]: there exists a [[basis]] of $V$ s with respect to which the [[matrix]] $\mathcal{M}_{\rho_{g}}$ is [[diagonal]] no matter what $g$ is. > [!proof]- Proof. ([[representation of finite abelian group consists of simultaneously diagonalizable operators]]) > Since $\rho \in \hom(G, \text{GL}(V))$, $\im \rho$ [[homomorphisms preserve structure|is abelian]] as a [[subgroup]] of $\text{GL}(V)$ and thus consists of [[commute|commuting]] [[linear operator|operators]]. Now, [[matrices of finite order in GL_n(C) are diagonalizable|since each]] $\rho_{g}$ is [[diagonalizable]], by [[diagonalizable matrices are simultaneously diagonalizable iff they mutually commute]] we have the result. ----- #### ---- #### References > [!backlink] [](group%20homomorphisms%20preserve%20structure.md)r Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```