I am studying a matrix computation course and have a homework question which I don't know how to tackle.
It introduced a key idea in the Coppersmith-Winograd Matrix Multiplication. Suppose n is even and define the following
\begin{gather} f(x) = \sum_{i=1}^{n/2} x_{2i-1}x_{2i},\space\space\space\space\space\space \forall x\in\mathbb{R^n} \end{gather}
Then it asked me to show that the inner product can be re-expressed as
\begin{gather} x^Ty = \sum_{i=1}^{n/2} (x_{2i-1}+y_{2i})(x_{2i}+y_{2i-1})-f(x)-f(y),\space\space\space\space\space\space \forall x,y\in\mathbb{R^n} \end{gather}
I tried working backwards by expanding the factorization above and then cancelling the f(x) and f(y)'s with the expanded terms, which left me with \begin{gather} \sum_{i=1}^{n/2} (x_{2i-1}y_{2i-1}+x_{2i}y_{2i}) \end{gather}
The above can probably be evaluated to $x^T$y but I don't know how. I know that $x^T$y = $\sum_{i=1}^{n} (x_{i}y_{i})$. Aside from that, I'm unsure what is the idea behind the f(x) equation, namely why does the summation end at n/2, and what are the $x_{2i}x_{2i}$ terms?
[full question][1] [1]: https://i.stack.imgur.com/nNZjs.png