----- > [!proposition] Proposition. ([[idempotent matrix iff diagonalizable and eigenvalues are all 0 or 1]]) > If [[matrix]] $P \in \mathbb{F}^{N \times N}$ is [[idempotent]] iff $P$ is [[diagonalizable]] with [[eigenvalue]]s all $0$ or $1$. > [!proof]- Proof. ([[idempotent matrix iff diagonalizable and eigenvalues are all 0 or 1]]) > Recall [[idempotent matrices are diagonalizable]]. Let $x$ be an [[eigenvalue]] of $P$, so that $Px=\lambda x$. Now multiply both sides by $P$ to get $P^{2}x=\lambda^{2}x$. But $P^{2}x=Px$ so $\lambda=\lambda^{2}$. Hence $\lambda \in \{ 0,1 \}$. \ Next, suppose $A$ is a [[diagonalizable]] [[matrix]] with [[eigenvalue]]s all $0$ or $1$. We write $A=VDV^{-1}$, where $D$ is [[diagonal matrix|diagonal]] with entries all $0$ or $1$. So $A^{2}=VDV V^{-1}DV^{-1}=VD^{2}V^{-1}$ but $D^{2}=D$. So $A^{2}=A$. ----- #### ---- #### 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 > ```