----- > [!proposition] Proposition. ([[eigenvalues of order-m matrix in GL_n(C) are mth roots of unity]]) > Let $A$ be a [[matrix]] of [[order of an element in a group|order]] $m$ in the [[general linear group]] over [[complex numbers]] . Each [[eigenvalue|eigenvalues]] of $A$ is an $m$ [[roots of unity]] : $\text{eig}(A) \subset \{ \omega_{n}^{\ell} : \ell =0,\dots,m-1 \}.$ > [!proof]- Proof. ([[eigenvalues of order-m matrix in GL_n(C) are mth roots of unity]]) > Let $v$ be an [[eigenvector]] of $A$ with [[eigenvector]] $\lambda$. We have $Av = \lambda v$ and by [[eigenvalues of matrix power]] $A^{n}v=\lambda^{n}v$. But $A^{n}=I$. thus $v=\lambda^{n}v$ implying $\lambda^{n}=1$. Thus $\lambda$ is an $n^{th}$ [[roots of unity|root of unity]]. ----- #### ---- #### 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 > ```