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Two matrices A and B are said to be similar if there is an invertible matrix P for which A = P BP^−1 . (i) Show that two similar matrices have the same spectrum. Is the converse true?

how to prove its converse part?

(ii) What about the eigen vectors? Are the same for two similar matrices?

Astro
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  • It is a natural question to ask and has been posted here before. The converse can fail if eigenvalues have repeats, for example a nilpotent matrix which is not the zero matrix. – hardmath Apr 10 '21 at 14:37
  • The summary of what is true: (a) Similar matrices have the same spectrum. (Show if $\lambda$ is an eigenvalue of $B$ for an eigenvector $x$, then $\lambda$ is an eigenvalue of $A=PBP^{-1}$ with an eigenvector $Px$.) (b) The converse is true if $A$ is an $n\times n$ matrix and it has all $n$ eigenvalues distinct. (Proof: Make up the matrix $P$ out of the eigenvectors of $A$ and you will get that $A$ is similar to a diagonal matrix with eigenvalues on the diagonal. $B$ is similar to the same matrix, therefore to $A$.) (c) The converse may be false if the eigenvalues are not distinct. –  Apr 10 '21 at 14:39

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