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 ...)

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- Jan 28th 2010, 11:51 AMSwlabrSymmetric matrix
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 ...) - Jan 28th 2010, 01:09 PMclic-clac
It is isn't it? This is a scalar (a 1-1 matrix).

Therefore it is equal to its transpose, and whenever is symmetric. - Jan 28th 2010, 01:19 PMSwlabr
- Jan 28th 2010, 06:07 PMNonCommAlg
- Jan 28th 2010, 10:52 PMSwlabr
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. - Jan 29th 2010, 12:33 AMOpalg