[[Eigendecomposition]] is a method of [[Decomposition]] for some [[Matrix]] $A$ in terms of some [[Matrix]] $\Lambda$ which is both [[Diagonal Matrix|Diagonal]] and [[Similar Matrices|Similar]] to $A$.
$\huge
A=Q \Lambda Q^{-1}
$
A [[Matrix]] $A\in M_{n\times n}$ is [[Eigendecomposition|Diagonalizable]] if there [[Existential Quantifier|Exists]] a [[Basis]]
of $\R^n$ that is composed of all [[Eigenvector|Eigenvectors]] of $A$. In other words, $A$ must have [[Eigenvector|Eigenvectors]] that [[Span]] $\R^n$.
>[!info]- Visualisation
>
For some [[Matrix]] $A$ and some [[Basis]] $Q$ composed from the [[Set]] of [[Eigenvector]] of $A$:
$\large
\begin{align}
\let Q &= \mat{ \vec \lambda_{1} & \dots \vec \lambda_{n}}\\
\Lambda &=Q^{-1}AQ =
\mat{
\lambda_{1} & 0 & \cdots \\0 & \lambda_{2} & \dots\\
\vdots & \vdots & \ddots
}\\
A&=Q \Lambda Q^{-1}
\end{align}
$
Breaking this down, [[Universal Quantifier|For Any]] [[Eigenvector]] $\vec \lambda_{i}$ of $A$, when using $A$ to [[Function|transform]] that [[Eigenvector]] (or any other [[Eigenvector]] [[Parallel]] to it), $\vec \lambda_{i}$ will be only be scaled by $\lambda$.
After the transformation, all [[Vector|Vectors]] on the line [[Span|spanned]] by $\vec \lambda_{i}$ will be scaled by $\lambda_{i}$.
The [[Matrix]] $Q$ is [[Composition|composed]] of all non-[[Parallel|parallel]] [[Eigenvector|Eigenvectors]] of $A$. This [[Matrix]] $Q$ is the [[Function|Transformation]] that moves the [[Standard Basis]] to the [[Eigenvector|Eigenvectors]] of $A$, consequently $Q^{-1}$ is the [[Function|Transformation]] that maps the [[Eigenvector|Eigenvectors]] of $A$ to the [[Standard Basis]].
$\Lambda$ is a [[Diagonal Matrix]] composed of the [[Eigenvalue|Eigenvalues]] of $A$.
The transformation first applies $Q^{-1}$, which is a [[Change of Basis Matrix]] that shifts the [[Eigenvector|Eigenvectors]] to the [[Standard Basis]]. After the $\Lambda$ matrix is applied which does a scaling of the standard basis by each eigen value. And finally, $Q$ is applied which brings the transformed [[Standard Basis]] [[Vector|Vectors]] to [[Eigenvector|Eigenvectors]] of $A$.
### Powers of a [[Matrix]]
This form can be used to calculate any $n$th power of $A$:
$\begin{align}
A^n &= (Q\Lambda Q^{-1}) \circ \dots \circ Q \Lambda Q^{-1} \\
&= Q \circ ( \Lambda \circ \dots \circ \Lambda) Q^{-1}\\
&= Q\Lambda ^nQ^{-1}
\end{align}
$
$
A^n=
\mat{ \lambda_{1} & \dots & \vec \lambda_{n} }
\mat{
\lambda_{1}^n & 0 & \cdots \\0 & \lambda_{2}^n & \dots\\
\vdots & \vdots & \ddots
}
\mat{ \lambda_{1} & \dots & \vec \lambda_{n} }^{-1}
$