Prove using the definition of eigenvalues that similar matrices have the same eigenvalues.
I have already shown this using the characteristic polynomial but I have no idea how to do it this way.
Any help would be greatly appreciated!
Prove using the definition of eigenvalues that similar matrices have the same eigenvalues.
I have already shown this using the characteristic polynomial but I have no idea how to do it this way.
Any help would be greatly appreciated!
Supose $\displaystyle A\sim B\Longrightarrow A=P^{-1}AP\Longleftrightarrow PAP^{-1}=B$ for some invertible matrix $\displaystyle P$.
Now let $\displaystyle Av=\lambda v\,\,and\,\,Pv=w\Longrightarrow P^{-1}w=v$, so then:
$\displaystyle Bw=PAP^{-1}w=PAv=P(\lambda v)=\lambda Pv=\lambda w\Longrightarrow \lambda$ is an eigenvalue of B
Tonio