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Suppose $A$ is some matrix and $S$ is a symmetric matrix. If I prove that $A$ is similar to $S$ (using a similarity transform), then are following statements true:

  1. $A$ is diagonalizable
  2. $A$ has distinct eigenvalues

Thanks in advance for helping out.

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If we're working over $\mathbb{R}$, then you use the spectral theorem to show that $S$ is similar to a diagonal matrix $D$, but then similarity is an equivalence relation, so that $A$ is also similar to $D$, and thus diagonalizable.

Now, I'm reading into (2) to mean that $A\in M_n(\mathbb{R})$ and that it has $n$ distinct eigenvalues, which isn't true. The identity is symmetric and has just one eigenvalue ignoring multiplicity.

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  • $\begingroup$ I forgot to mention that we are only dealing with real matrices here. So, we are working over $\mathbb R$, as you assumed. Also, it seems to me that your answer is correct. I will accept it soon. $\endgroup$ – Sandeep Jul 21 '17 at 23:12

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