Let the sets of eigenvalues and eigenvectors of a matrix $B$ be $\left\{\lambda_k \mid 1 \leq k \leq n\right\}$ and $\left\{v_k \mid 1 \leq k \leq n\right\}$, respectively. For any invertible matrix $P$, the sets of eigenvalues and eigenvectors of the matrix $A$, where $B=P^{-1} A P$, respectively, are
- $\left\{\lambda_k \operatorname{det}(A) \mid 1 \leq k \leq n\right\}$ and $\left\{P v_k \mid 1 \leq k \leq n\right\}$
- $\left\{\lambda_k \mid 1 \leq k \leq n\right\}$ and $\left\{v_k \mid 1 \leq k \leq n\right\}$
- $\left\{\lambda_k \mid 1 \leq k \leq n\right\}$ and $\left\{P v_k \mid 1 \leq k \leq n\right\}$
- $\left\{\lambda_k \mid 1 \leq k \leq n\right\}$ and $\left\{P^{-1} v_k \mid 1 \leq k \leq n\right\}$
