Similar Matrices Have the Same Eigenvalues
Theorem
If two matrices $A,B$ are similar, they have the same eigenvalues.
$$ \det (A - \lambda I) = \det (B - \lambda I) $$
In this case, $\lambda$ is an eigenvalue of $A, B$.
Description
Having the same eigenvalues means that the characteristic equations are the same.
Proof
$$ \begin{align*} \det (A - \lambda I ) =& \det ( P^{-1} (B - \lambda I ) P ) \\ =& \det P^{-1} \det (B - \lambda I) \det P \\ =& \det P^{-1} \det P \det (B - \lambda I) \\ =& \det I \det (B - \lambda I) \\ =& \det (B - \lambda I) \end{align*} $$
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