----- > [!proposition] Proposition. ([[eigenvalues of adjoint]]) > Let $T$ be a [[linear operator]] on a [[finite-dimensional vector spaces MOC|finite-dimensional]] [[vector space]] $V$ over $\mathbb{R}$ or $\mathbb{C}$. [[Gram-Schmidt Procedure|Fix an]] [[orthonormal basis]] of $V$ and let $A$ denote the [[matrix]] of $T$ with respect to it. Then: > 1. The [[eigenvalue]]s of $T^{\dagger}$ are the [[complex conjugate]]s of the [[eigenvalue]]s of $T$; > 2. The [[eigenvalue]]s of $A'$ are the [[complex conjugate]]s of the [[eigenvalue]]s of $A$. > [!proof]- Proof. ([[eigenvalues of adjoint]]) > $T^{\dagger}$ and $A$ will have the same [[eigenvalue]]s, so it suffices to compute the [[eigenvalue]]s of $A$ [[eigenvalue iff solves characteristic polynomial|using its characteristic polynomial]]. Recalling that [[determinant of matrix equals determinant of transpose]] we have $\begin{align} \det (A - \lambda I)= & \det \big( (A-\lambda I)'\big) \\ = & \det (A' - \lambda I') \\ = & \det (A' - \lambda I), \end{align}$ as required. ----- #### ---- #### 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 > ```