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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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Unless you're dealing with multiple $S$'s, using differently shaped brackets works, e.g. $[\vec{x},\vec{y}]=\vec{x}^\mathrm{T}S\vec{y}$ or $\left(\vec x,\vec y\right)=\vec{x}^\mathrm{T}S\vec y$. I don't know if there is a particular "standard" way. – Jonas Meyer May 5 '11 at 17:24
Actually, your use of $\langle \vec{x}, \vec{y} \rangle_S$ is quite standard (anyway, you shouldn't use any notation without explaining or defining it). I'd refrain from using Jonas's suggestion $[\vec{x},\vec[y]]$ because square brackets are usually reserved for Lie brackets. – t.b. May 5 '11 at 20:10
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. – Lagerbaer May 5 '11 at 20:15

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