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?
2026-03-26 22:19:31.1774563571
Real symmetric matrix and real eigenvalues
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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!