> [!NOTE] Theorem > Let $A,B \in \text{Mat}_{nn}(\mathbb{R})$ be [[Similar Real Square Matrices|similar real square matrices]]. Then $A,B$ have the same [[Eigenpair of Real Square Matrix|eigenvalues]]. **Proof**: Since $A,B$ are similar, there exists invertible matrix $P\in \text{Mat}_{nn}$ such that $B=P^{-1}AP.$ Then $A=PBP^{-1}.$ We have that $\begin{aligned} \operatorname{det}\left(A-\lambda I_n\right) & =\operatorname{det}\left(P B P^{-1}-\lambda P P^{-1}\right) \\ & =\operatorname{det}\left(P\left(B-\lambda I_n\right) P^{-1}\right) \\ & =\operatorname{det}(P) \operatorname{det}\left(B-\lambda I_n\right) \operatorname{det}\left(P^{-1}\right) \\ & =\operatorname{det}(P) \operatorname{det}\left(P^{-1}\right) \operatorname{det}\left(B-\lambda I_n\right) \\ & =\operatorname{det}\left(B-\lambda I_n\right) \end{aligned}$