The change of basis matrix is orthogonal, which implies that its determinant is . It will be +1 if the basis change is orientation-preserving, and –1 if it is orientation-reversing. The formula is correct.
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!