Once again, this is from a past qualifying exam I am trying to work on.
Here is the problem.
True or False? Let $A$ be a real $n\times n$ matrix such that $AA^T=A^TA$ and all eigenvalues of $A$ are real. Then $A$ must be symmetric.
Attempt. Well it looks as if this is related to the real spectral theorem. I know that any real symmetric matrix is diagonalizable and must have real eigenvalues. This is sort of a converse to that theorem right? I tried a few $2\times 2$ examples and they all point to the above statement being True. Am I right?. But I am unable to see a way to prove. We actually haven't learnt the spectral theorem nor is it on the qual syllabus. So I'm kind of unsure on this. Can you help?
Thank you in advance.