Suppose A and B are similar matrices. Show that A and B have the same eigenvalues with the same geometric multiplicities.
Similar matrices: Suppose A and B are n×n matrices over R or C. We say A and B are similar, or that A is similar to B, if there exists a matrix P such that B=P−1AP.
B=P−1AP ⟺ PBP−1=A. If Av=λv, then PBP−1v=λv ⟹ BP−1v=λP−1v. so, if v is an eigenvector of A, with eigenvalue λ, then P−1v is an eigenvector of B with the same eigenvalue. So, every eigenvalue of A is an eigenvalue of B and since you can interchange the roles of A and B in the previous calculations, every eigenvalue of B is an eigenvalue of A too. Hence, A and B have the same eigenvalues.
Geometrically, in fact, also v and P−1v are the same vector, written in different coordinate systems. Geometrically, in fact, also A and B are matrices associated to the same endomorphism. So, they have the same eigenvalues, eigenvectors and geometric multiplicities.