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# Basic inner product question watch

1. Either my memory's gone or this is something I've not quite grasped for some reason:

I get that

Now I'm looking at some notes on orthogonal matrices and it says that

Question 1) This is only working because the dot product is defined to be , right? So that it wouldn't work for any other inner products?

Question 2) But using the above property, surely even if I define my inner product to be the dot product, I should get:

2. The dot product is the inner product with respect to the identity matrix I. A popular choice, since the columns of I give the standard orthonormal basis. To be clear, you may want to write <x,y>R = xTRy (where R is symmetric positive definite), making it clear that an inner product depends on a choice of matrix to take the inner product with represent to.

So for Qu2, your problem arises from an ambiguity over what A is representing.
3. (Original post by Glutamic Acid)
The dot product is the inner product with respect to the identity matrix I. A popular choice, since the columns of I give the standard orthonormal basis. To be clear, you may want to write <x,y>R = xTRy (where R is symmetric positive definite), making it clear that an inner product depends on a choice of matrix to take the inner product with represent to.

So for Qu2, your problem arises from an ambiguity over what A is representing.
Ah ok, thanks.

In general, how do you know what R is for a given inner product space?
4. If your basis is x1, x2, ..., x_n, then I think you'll find that . (You might need to fiddle a little, although I don't think so).

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Updated: April 14, 2011
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