Similar matrices have the same eigenvalues with the same geometric multiplicity

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=P1AP.

Answer

B=P1AP  PBP1=A. If Av=λv, then PBP1v=λv  BP1v=λP1v. so, if v is an eigenvector of A, with eigenvalue λ, then P1v 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 P1v 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.

Attribution
Source : Link , Question Author : Community , Answer Author : Agustí Roig

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