Proving similar matrices if diagonal have same entries which are eigenvalues

[Math_Fortnite_Lover]

I know that since (A-tI)x+0, the diagonal matrix has all the eigenvalues.

I also have proved in my homework that similar matrices have the same eigenvalues. But I am stuck on how to show that in a proof way. Any help would be appreciated?

 

[blamocur]

I believe similar matrices have identical characteristic polynomials, which means that their sets of eigenvectors are identical too.

 

[Math_Fortnite_Lover]

I believe similar matrices have identical characteristic polynomials, which means that their sets of eigenvectors are identical too.

Thank you. I was thinking this too but just am not sure how to write it in proof form.