When is the matrix of eigenvectors of a complex symmetric matrix orthogonal?

Question

Given a complex symmetric matrix $A=A^\top$ with a matrix of eigenvectors $C$ (which have distinct non zero eigenvalues) it can be shown that $C^\top C=I$ and that $C^\top A C=D$ where $D$ is a diagonal matrix of the eigenvalues.

I want to know how to show that further to the above $C^\top C= C C^\top =I$. And if this is not always true when it will be true.

Answer 1

Based on the condition that $C^\top C=I$ it follows that since $C$ is a square matrix $C C^\top=I$ see:

If $AB = I$ then $BA = I$