I'm wondering how to tackle down and calculate the following equation efficiently:
$\sum_{i=1}^m \sum_{j=m+1}^n x_i x_j (\phi_i^T \phi_j)$
Am I allowed to do this?
$(\sum_i^m x_i) (\sum_j^n x_j) (\sum_i^m \sum_j^n(\phi_i^T \phi_j))$
And if so, how do I calculate $\sum_i^m \sum_j^n(\phi_i^T \phi_j)$?
What's exactly the difference between $\sum_i^m \sum_j^n(\phi_i^T \phi_j)$ and $\sum_i^m \sum_j^n(\phi_i \phi_j)$ in this case?
Note that $x$ and $\phi$ are $n\times 1$ vectors
It is not very clear what you want to achieve, and your notation could do with some clarification. I did my best to decipher what you are asking.
You are not allowed to do that series manipulation. Since $x_i$ is constant with respect to the index $j$, the best you can do is $\sum_{i=1}^m x_i\sum_{j=1}^m x_j(\phi_i^T\phi_j)$. The rest must remain under the same series.
In general, you cannot split a series in the following way: $\sum ab \to \sum a\sum b$. Think of the difference between the sum $x_1y_1+x_2y_2+x_3y_3$ and $(x_1+x_2+x_3)(y_1+y_2+y_3)$.
If, as you say, $\phi_i$ and $\phi_j$ are $m\times1$ and $n\times 1$ vectors respectively (i.e., column vectors), then $\phi_i^T\phi_j$ is only defined when $m=n$, and it is equal to their dot product $\langle\phi_i,\phi_j\rangle$. On the other hand, $\phi_i\phi_j$ is not even defined.
Edit: OP, you have edited your question so that $x_i$ and $x_j$ are also vectors. In this case, $x_ix_j(\phi_i^T\phi_j)$ does not make any sense.