# Real symmetric matrix and real eigenvalues

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?

• Definitely not. You can probably think of a $2$ by $2$ case very easily. – MathIsLife12 Jan 30 at 5:01
• Thanks, [1 1; 1 0]. I think prove that it's wrong – ChaosPredictor Jan 30 at 5:06

## 1 Answer

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!