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I have been wondering if the following dot product definition for the $n$-coordinate vectors $a$ and $b$ has a name: $$<a\backslash b> = \sum_{i=1}^{n} a_i*b_{n-i+1},$$ rather than the classical dot product: $$<a\backslash b> = \sum_{i=1}^n a_i*b_{i}.$$ Did you already seen it use somewhere?

Thanks a lot.

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Should it be $$<a\backslash b> = \sum_{i=1}^n a_i*b_{n-i+1}?$$ –  yoyostein Jul 19 '12 at 8:57
    
Yes, my mistake. Corrected, thanks. –  wwjoze Jul 19 '12 at 9:00
    
you could write as $\langle a | X b \rangle$, where $X$ is antidiagonal unit matrix... maybe you can interpret this as scalar product in a space with metric $X$... –  draks ... Jul 19 '12 at 9:06

2 Answers 2

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You can diagonalize this bilinear form so that it becomes the sum of $p$ squares minus the sum of $q$ squares where $p$ and $q$ are $n/2$ rounded up and down, respectively. This is called a quadratic form of signature $(p,q)$, and the space in which this norm is the inner product is denoted $\mathbb R^{p,q}$. For example, see http://en.wikipedia.org/wiki/Indefinite_orthogonal_group

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After a reindexing, I would call what you have a (discrete) circular convolution.

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