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As I understood all eigenvalues of real symmetric matrix are real. But is it true that any real matrix with all real eigenvalues is symmetric?

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  • $\begingroup$ Definitely not. You can probably think of a $2$ by $2$ case very easily. $\endgroup$ – MathIsLife12 Jan 30 at 5:01
  • $\begingroup$ Thanks, [1 1; 1 0]. I think prove that it's wrong $\endgroup$ – ChaosPredictor Jan 30 at 5:06
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No! Take any non zero nilpotent matrix with real entries!

For example, $$\begin{pmatrix} 0&1\\0&0\end{pmatrix}$$


If any real matrix with all real eigenvalues is symmetric, then we have a conclusion: $$\text{any real matrix with all real eigenvalues is diagonalizable!}$$ which is in general false!

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