Ytukyg

Reading an article about clustering, I came across this equality:

It's basically a sum over $ mu_{i,j}(1-cos(x_i, p_j))$ (where $ cos(x_i, p_j) $ is the cosinus between two vectors in $mathbb{R}^n$). And then suddenly some magic happens and the formula in the right appears. Can somebody explain?

Note: I understand that the parenthesis was expanded (each term got multiplied by $ mu_{i,j}$). But what is that huge dot product and why does it have this form?

Remember that the inner product is bilinear: for three vectors $u,v,w$ and scalars $alpha,beta$,
$$
langle alpha u, beta vrangle = alphabetalangle u, vrangle, qquad langle u, v+wrangle = langle u, vrangle+langle u, wrangle
$$
What happened above is exactly what follows when applying these rules: for every $i,j$
$$
frac{langle x_i, p_jrangle}{lVert x_irVertlVert p_jrVert} = leftlangle frac{x_i}{lVert x_irVert}, frac{p_j}{lVert p_jrVert}rightrangle tag{1}
$$
since $lVert x_irVert,lVert p_jrVertinmathbb{R}$ are scalars (the brackets became "bigger" just for style considerations, like parentheses would be drawn bigger when what's inside looks "large").

Then, each $mu_{ij}$ is also a scalar, and you have a sum, so
$$
sum_{i} mu_{ij}leftlangle frac{x_i}{lVert x_irVert}, frac{p_j}{lVert p_jrVert}rightrangle
= sum_{i} leftlangle mu_{ij}frac{x_i}{lVert x_irVert}, frac{p_j}{lVert p_jrVert}rightrangle
= leftlangle sum_{i}mu_{ij}frac{x_i}{lVert x_irVert}, frac{p_j}{lVert p_jrVert}rightrangle tag{2}
$$
Note that it does make sense: $sum_{i}mu_{ij}frac{x_i}{lVert x_irVert}$ is a vector, and so is $frac{p_j}{lVert p_jrVert}$. So it's indeed the inner product between two vectors.