Let by a symmetric matrix. Then is it true in general that for and column vectors?
If not, when is it true?
(I'm trying to show that forms an inner product, but as far as I can see ...)
Hmm. Well, I've been asked to prove that the assignment,
, is an inner product, where and the are the basis vectors for , the tangent space of a manifold at a point .
So I need to prove that it is positive definite, but I can't see why this holds. I suppose the point to use would be that elements on the diagonal are all (strictly) positive, but I don't see how to use this.