---- - Let $\ff$ denote $\rr$ or $\cc$. > [!definition] Definition. ([[normal matrix]]) > A [[matrix]] $A \in \ff^{n \times n}$ is called **normal** if it commutes with its [[conjugate transpose]]; i.e., if $A^{\dagger}A = A A ^{\dagger}.$ > > [!basicproperties] Important. (A [[normal operator]]'s [[matrix]] is [[normal matrix|normal]]) > Let [[inner product space]] be an [[inner product space|inner product space]]. Let $T \in$ [[vector space of operators on a vector space]] be a [[normal operator]] and $\{ w_{j} \}_{j=1}^{n}$ be a [[basis]] for $V$. By definition of [[matrix product]], $\begin{align} \MM(T, \{ w_{j} \}_{j=1}^{n})\MM(T^{\dagger}, \{ w_{j} \}_{j=1}^{n})= & \MM(TT^{\dagger}, \{ w_{j} \}_{j=1}^{n}) \\ = & \MM(T^{\dagger} T, \{ w_{j} \}_{j=1}^{n}) \\ = & \MM(T^{\dagger}, \{ w_{j} \}_{j=1}^{n})\MM(T, \{ w_{j} \}_{j=1}^{n}), \end{align}$ so we see that $\MM(T, \{ w_{j} \}_{j=1}^{n})$ is a [[normal matrix]]. ---- #### ---- #### 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 > ```