Matrix Exponential

Definition

$e^{\mathbf{A}}:={\sum }_{k=0}^{\infty }\frac{1}{k!}{\mathbf{A}}^k$ where ${\mathbf{A}}^0=\mathbf{I}$

If $\mathbf{A}=\mathbf{S\Lambda }{\mathbf{S}}^{-1}$ is Diagonalizable Matrix, $e^{\mathbf{A}}:=\mathbf{S}{e}^{\mathbf{\Lambda }}{\mathbf{S}}^{-1}$

$a^{\mathbf{A}}:={\sum }_{k=0}^{\infty }\frac{(\ln a{)}^k}{k!}{\mathbf{A}}^k$

If $\mathbf{A}=\mathbf{S\Lambda }{\mathbf{S}}^{-1}$ is Diagonalizable Matrix, $a^{\mathbf{A}}:=\mathbf{S}{e}^{\ln a\mathbf{\Lambda }}{\mathbf{S}}^{-1}$

Differential of $e^{\mathbf{A}t}$

$\frac{d}{dt}{e}^{t\mathbf{A}}=\mathbf{A}{e}^{t\mathbf{A}}$