Examples:: [[diagonalizable#^08d06c|basic example]] Nonexamples:: *[[Nonexamples]]* Constructions:: *[[Constructions|Used in the construction of...]]* Specializations:: *[[Specializations]]* Generalizations:: *[[Generalizations]]* Justifications and Intuition:: *[[Justifications and Intuition]]* Properties:: [[eigendecomposition of a matrix]] Sufficiencies:: [[enough eigenvalues implies diagonalizability]] Equivalences:: [[diagonalizable iff admits an eigenbasis]], [[diagonalizable iff V is a direct sum of one-dimensional invariant subspaces]], [[diagonalizable iff V a direct sum of eigenspaces]], [[diagonalizable iff eigenspace dimensions sum to supspace dimension]], [[diagonalizable iff arbitrary matrix similar to diagonal matrix]], [[diagonalizable iff minimal polynomial is product of distinct factors]] ---- > [!definition] Definition. ([[diagonalizable]]) > A [[linear map]] $T \in$[[vector space of operators on a vector space]] is said to be **diagonalizable** or **semisimple** if $T$ has a [[diagonal matrix]] with respect to some [[basis]] of $V$. > [!basicexample]- > Define $T \in \endo(\rr^2)$ by $T(x,y)=(41x+7y,-20x+74y)$. With respect to the standard basis of $\rr^2$ the [[matrix]] of $T$ is $\begin{pmatrix} 41 & 7 \\ -20 & -74 \end{pmatrix},$ which is clearly not [[diagonal matrix|diagonal]]. However, with respect to the [[basis]] $(1,4),(7,5)$ the [[matrix]] of $T$ is $\begin{pmatrix} 69 & 0 \\ 0 & 46 \end{pmatrix},$ hence $T$ is [[diagonalizable]]. ^08d06c > [!equivalence]- Equivalences. > - [[diagonalizable iff admits an eigenbasis]] > - [[diagonalizable iff V is a direct sum of one-dimensional invariant subspaces]] > - [[diagonalizable iff V a direct sum of eigenspaces]] > - [[diagonalizable iff eigenspace dimensions sum to supspace dimension]] > - [[diagonalizable iff arbitrary matrix similar to diagonal matrix]] > [!definition] Definition. ([[diagonalizable]] [[matrix]]) > Let $A$ be an $n \times n$ [[matrix]]. We say $A$ is **diagonalizable** if the [[linear map]] $\mathbb{R}^n \xrightarrow{T_A} \mathbb{R}^n \text{ defined by } T_A(\vec x) = A \vec x $ is **diagonalizable**. \ So, $A$ is **diagonalizable** if there exists an [[inverse matrix|invertible]] [[matrix]] $S$ (possibly with [[complex]] entries) and [[diagonal matrix]] $D = \text{diag}(\lambda_1, \dots, \lambda_n)$, $\lambda_i \in \mathbb{C}$ for $1 \leq i \leq n$, such that $A = SDS^{-1}.$ That is to say: a square [[matrix]] is said to be [[diagonalizable]] if it is [[similar]] to a [[diagonal 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 > ```