⇒To start with, according to the similarity theorem, they have the same eigenvalues so is the multiplicity.
⇐If two matrices have the same eigenvalues and multiplicity, that implies they are similar to the same diagonalizable matrix D and by equivalent relation, they are similar to each other.
However for the second part, since we are not sure the dim(Eλn)=multiplicity for λn, how are we making sure that the two matrices are diagonalizable. If diagonalizability is not assured, my second part is wrong.
Could you offer me any hint?
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