As as part of my course in Lie groups, I need some help to determine the signature of the form $B(x,y) = $$ \sum_{i=1}^{n}\ x_iy_{n+1-i}$
More than anything, I would like to understand how to determine a signature of such forms, rather than the plain answer.
thanks in advance!
The associated quadratic form has a slightly different nature depending on the parity of $n$. For $n$ odd, it is $x_1x_n+x_2x_{n-1}+\ldots+x_{{n+1\over 2}}^2$, while for $n$ even there is no pure square term.
The summands $x_ix_j$ with $i\not=j$ are "hyperbolic pairs", and have signatures $(1,-1)$. The pure square term, for $n$ odd, has signature $(1,0)$. Add them up.
(There are many supporting exercises one could do here... in the direction of "geometric algebra", and Sylvester's Inertia Theorem, but I don't know your context.)