Orthogonality of Eigenvectors corresponding to different type of matrices

Question

I've learnt the amazing eigensystem of real symmetric matrices has the 2 following properties:

1. The eigenvalues are real number.
2. The eigenvectors could be chosen to be orthogonal when we diagonalize it.

As I keep learning through 18.06 to Unit 3 Exam Review, prof. Strang noted that:

A matrix has orthogonal eigenvectors exactly when $AA^T$ = $A^TA$; i.e. when A commutes with its transpose. This is true of symmetric, skew symmetric and orthogonal matrices.

Does anybody know how can I prove the theorem above starting from $AA^T$ = $A^TA$? Is there any useful proposition I can use?

It's my first time to ask question on this forum. Any suggestion or hint would be appreciated. Thanks.