Suppose I have a matrix w.r.t some basis .

Further suppose that this matrix undergoes a basis change to where each is orthonormal to another (ie. is an orthonormal basis). Call this matrix (which has columns ).

Let the change of basis matrix be .

Is it true that ?

Is it also true that ?

(P.S I've done a question and this is one of the facts I used. I'm wondering if it really is a fact!).

Thanks in advance!