[[Noteworthy Uses]]:: *[[Noteworthy Uses]]* [[Proved By]]:: *[[Proved By|Crucial Dependencies]]* Intuition:: *[[Intuition]]* Examples:: [[spectral decomposition example]] Specializations:: [[spectral decomposition of a matrix]] Generalizations:: *[[Generalizations]]* ---- Here, $\ff$ denotes $\rr$ or $\cc$. > [!theorem] Theorem. ([[eigendecomposition of a matrix]]) > Then there exists an [[inverse matrix|invertible]] [[matrix]] $S \in \ff^{n \times n}$ such that $A = SDS^{-1},$ where $S$ consists of [[eigenvector]]s of $A$ and $D$ is [[diagonal]] with entries corresponding to the associated [[eigenvalue]]s. > [!proof]- Proof. ([[eigendecomposition of a matrix]]) > **Remark**. It may seem like this proof is losing generality by only considering $\cc ^{n}$ and $\rr ^{n}$. However it is concerned *only* with the decomposition of arbitrary **matrices** (*that may not even be interpreted in association to any [[linear map]] at all!*), and uses as a convenience **linear maps** from $\ff ^{ n}$ to $\ff ^{n}$ as a means to prove the result. \ -Denote by $\mathcal{B}$ the standard [[basis]] of $\ff ^{n}$. -Define $T \in \endo(\ff^n)$ to be the [[linear map]] given by $T(x)=Ax$ (i.e., the [[linear map]] whose [[matrix]] w.r.t. $\mathcal{B}$ is $A$; $A = \MM(T, \mathcal{B})$ ) [^1]. -Using that $T$ is [[diagonalizable]], [[diagonalizable iff admits an eigenbasis|fix an eigenbasis]] $\mathcal{E}$ of $\ff^n$ and a [[diagonal matrix]] $D=\MM(T,\mathcal{E})$. \ Let $S$ be the [[transition matrix]] from $\mathcal{E}$ to $\mathcal{B}$— i.e., the [[matrix]] whose columns correspond to the members ([[eigenvector]]s) of $\mathcal{E}$. In particular, $S$ is the [[matrix]] of the [[identity map]] with respect to these [[basis|bases]]: $S = \MM(I, \mathcal{E}, \mathcal{B}),$ [[matrix of the identity w.r.t. two bases|note that]] $S^{-1} = \MM(I, \mathcal{B}, \mathcal{E}).$ Because $T=ITI^{-1}$ and by definition [[matrix product|the product of matrices is the matrix of the product of linear maps]], we have $\begin{align} A = & \MM(T, \mathcal{B} \to \mathcal{B}) \\ = & \MM(ITI^{-1}, \mathcal{B} \to \mathcal{E} \to \mathcal{B}) \\ = & \MM(I, \mathcal{E}, \mathcal{B}) \MM(T, \mathcal{E})\MM(I, \mathcal{B}, \mathcal{E})) \\ = & SDS^{-1}. \end{align}$ $\qedin$ $\qedin$ [^1]: In part since if $\{ e_{j} \}_{j=1}^{n}$ denotes the standard [[basis]] we have $T(e_{1}) = Ae_{1}= (\text{first column of }A)$, $Ae_{1}= (\text{second column of }A)$, etc... ---- #### ----- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as 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 > ```