Decomposition of Matrix Exponential

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Let $\mathbf A$ be a square matrix of order $m$ for some $m \in \Z_{\ge 1}$.

Let $t \in \R$ be a real number.

Let $e^{\mathbf A t}$ denote the matrix exponential of $\mathbf A$.

Let $\mathbf P$ be a non-singular matrix of order $m$.


$e^{\mathbf P \mathbf A \mathbf P^{-1} } = \mathbf P e^{\mathbf A} \mathbf P^{-1}$


$\paren {\mathbf P \mathbf A \mathbf P^{-1} }^n = \mathbf P \mathbf A^n \mathbf P^{-1}$ by induction.