So, I tried to prove this question in two parts (A) if A is a symmetric matrix then it is orthogonal (B) since A is orthogonal, its row vectors must form an arthonormal set. But I don't think it is possible to prove part (A).
Anyway here's my proof so far:
If A is symmetric then . A is an orthogonal matrix if (since and are symmetric and commute). And must have an inverse .
Let denote the rows of ; then are the columns of .
By matrix multipication:
Therefore rows of A form an orthonormal set.
My proof is proof is probably not correct. Can anyone show me how to prove this?