\[ \mbox{Similar Matrix} \] For invertable matrix $\mathbf{A}$, if there exist matrix $\mathbf{P}$ such that \[ \mathbf{A} = \mathbf{P} \mathbf{B} \mathbf{P}^{-1} \] then $\mathbf{A}$ is similar to $\mathbf{B}$

\[ \mbox{Proposition} \] Let $\mathbf{A}$ be an $n \times n$ matrix, $\mathbf{P}$ is non-singular matrix $n \times n$ matrix, and $\mathbf{B} = \mathbf{P} \mathbf{A} \mathbf{P}^{-1}$. Then the matrices have the same characteristic polynomials.

Proof. The characteristic polynomials of $\mathbf{A}$ and $\mathbf{B}$ are given as \begin{equation} \begin{aligned} \det (\mathbf{A} - \lambda \mathbf{I}) & = \det (\mathbf{P} \mathbf{B} \mathbf{P}^{-1} - \lambda \mathbf{I}) \\ & = \det (\mathbf{P} \mathbf{B} \mathbf{P}^{-1} - \lambda \mathbf{P} \mathbf{P}^{-1}) \\ & = \det (\mathbf{P} \mathbf{B} \mathbf{P}^{-1} - \mathbf{P} \lambda \mathbf{P}^{-1}) \\ & = \det (\mathbf{P} \mathbf{B} \mathbf{P}^{-1} - \mathbf{P} \lambda \mathbf{I} \mathbf{P}^{-1}) \\ & = \det (\mathbf{P} (\mathbf{B} - \lambda \mathbf{I}) \mathbf{P}^{-1}) \\ & = \det (\mathbf{P}) \det(\mathbf{B} - \lambda \mathbf{I}) \det (\mathbf{P}^{-1}) \\ & = \det(\mathbf{B} - \lambda \mathbf{I}) \\ \end{aligned} \end{equation}