Perpendicular eigenvectors and symmetry

Question

Assume, that matrix $A \in \mathbb{R}^{n \times n}$ has $n$ eigenvectors, which are all perpendicular with respect to one another. Show that $A$ is symmetric.

This was an extra question on an exam and Im curious on how one would show this?

Answer 1

Let your matrix be denoted by $A$. If your eigenvectors are perpendicular to each other, divide them to their norms and then you will get n unit orthogonal vectors which are also your eigenvectors. Then these eigenvectors can create orthogonal matrix $Q$ and we know $Q^T=Q^{-1}$. After using matrix diagonalization, you will see that $A^T=(Q^{-1}DQ)^T=(Q^TDQ)^T=Q^TDQ=Q^{-1}DQ=A$.