----- > [!proposition] Proposition. ([[matrices of finite order in GL_n(C) are diagonalizable]]) > Let $A$ in the [[general linear group]] have [[order of an element in a group|order]] $m$. Then $A$ is [[diagonalizable]]. > [!proof]- Proof. ([[matrices of finite order in GL_n(C) are diagonalizable]]) > Consider the [[cyclic group]] $\langle A \rangle$. The [[inclusion map|inclusion]] of $\langle A \rangle$ into $\text{GL}_{n}(\mathbb{C})$ is a [[group representation|matrix representation]] of $\langle A \rangle$. > > Consider the [[cyclic group]] $\langle T \rangle=: G$. The inclusion of $T$ into $\text{GL}(V)$ is a [[group representation|representation]] $\rho$ of $T$. Give $V$ any [[inner product]] $\langle \cdot,\cdot \rangle$, and [[group-averaged Hermitian form|use it to endow]] $V$ with a $G$-[[group-invariant function|invariant]] [[inner product]] $\langle \langle \cdot,\cdot \rangle \rangle$ by [[averaging over a group|averaging wrt]] $\rho$. > > We now induct on $\dim V$. If $\dim V=1$ then the result is trivial. Suppose the result holds when $\dim V = n>1$. [[over the complex numbers, every operator has an upper-triangular matrix|Because the ground]] [[field]] is [[complex numbers]], $T$ has an [[eigenvalue]] $\lambda$ with [[eigenspace]] $E_{\lambda}$ that is $T$-[[invariant subspace|invariant]], or equivalently $\rho_{T}$-[[group-invariant subspace|invariant]] ($T=\rho_{T}$). Now, > > - If $E_{\lambda}=V$ we are done since of course $T$ restricted to $E_{\lambda}$ is [[diagonalizable]] (being a [[linear subspace]] of [[eigenvector|eigenvectors]], any [[basis]] of $E_{\lambda}$ is an [[eigenbasis]] of $E_{\lambda}$). > - Else $\dim E_{\lambda} < n$ and so $E_{\lambda}$ [[group-invariant subspace admits group-invariant complement over C|admits a]] $G$-invariant [[complement of a linear subspace|complement]] $E_{\lambda}'$. $T=\rho_{T}$ acts on both of these spaces and hence restricts to both of them; by the induction hypothesis it is [[diagonalizable]] restricted each of them, and in turn is [[diagonalizable]] on their [[direct sum of vector spaces]] $V=E_{\lambda} \oplus E_{\lambda}'$. (Because the [[matrix]] of $T=\rho_{T}$ with respect the concatenated bases is always block [[block matrix|diagonal]]... and in this case each block is itself [[diagonal]]). > > ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY 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 > ```