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Maybe not precisely a math question, but certainly related, and apparently there even is a notation tag :)

We can think of the standard vector scalar product, $ \langle \vec{x}, \vec{y} \rangle := \sum_i x_i y_i $ as being a special case of matrix-vector multiplication, i.e. $ \langle \vec{x}, \vec{y} \rangle = \vec{x}^T I \vec{y}$ with $I$ the unit matrix.

I can then define a generalized dot product with respect to another matrix $S$ via $ \langle \vec{x}, \vec{y} \rangle_S := \vec{x}^T S \vec{y}$ If $S$ is symmetric positive definite, the map $\langle \cdot, \cdot \rangle$ together with the vector space I'm looking at is still an Euclidean space, i.e. it still behaves like a scalar product.

Is there a standard way in the mathematical literature for the notation of such a generalized scalar product? I fear that my way of writing it down might be a bit cumbersome and might be a source of confusion for the reader, who could, when reading it, mistake it for a typesetting error where the matrix $S$ got mistakenly subscripted?

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    Yeah, I have, of course, defined it. But a willing test subject for my write-up was startled by it once or twice so I thought I should seek advise.2011-05-05

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