---- > [!definition] Definition. ([[simultaneously diagonalizable]]) > A set of [[matrix|matrices]] $A_{1}, A_{2},\dots,$ is **simultaneously [[diagonalizable]]** if there exists a single [[inverse matrix|invertible matrix]] $P$ such that $P ^{-1} A_{k} P$ is [[diagonal matrix|diagonal]] for every $A_{k}$ in the set. > > Similar notions hold for (strictly) triangularizable, etc. > $V$ f.d. A subspace (?) of [[linear operator|linear operators]] $\mathcal{L} \subset \text{End}(V)$ is [[simultaneously diagonalizable]] if there exists a [[basis]] $\{ v_{1},\dots, v_{n} \}$ of $V$ which is a common [[eigenbasis]] for each $T \in \mathcal{L}$. > Each common [[eigenvector]] $v_{i}$ induces a [[linear functional]] $\lambda_{i} : \text{End}(V) \to \mathbb{F}$ assigning $T \mapsto \lambda_{i}(T)$, where $\lambda_{i}(T)$ is the [[eigenvalue]] of $T$ corresponding to $v_{i}$: $Tv_{i}=\lambda_{i}(T)v_{i}$. > This yields a direct sum decomposition of $V$ as follows: define for $\lambda_{i}$, $i \in [n]$, the 'simultaneous $\lambda_{i}$-eigenspace': $V_{\lambda_{i}}:=\{ v \in V: Tv=\lambda_{i}(T)v \text{ for all } T \in \mathcal{L}\}$ Then, since the $V_{\lambda_{i}}$ intersect trivially and sum to $V$, we have $V=\bigoplus_{i=1}^{n}V_{\lambda_{i}}.$ ---- #### ---- #### 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 > ```