Convert a general equation system to matrix form

Question

I have an equation, where $x$ is a vector of variables and the rest (vector $a$, vector $b$, matrix $M$ and $c_1, c_2, c_3$) are parameters:

$a_i - x_i + c_1\sum_{j=1}^{n} m_{ij} x_j - c_2 \sum_{j=1}^{n} x_j + c_3 (b_i \sum_{j=1}^{n} x_j) = 0, \forall i=1,...,n$

This is my thinking on making this in vector form ($U$ is the $n \times n$ matrix of ones):

$a - x + c_1 M x - c_2 Ux + $ ?? $=0$

I am struggling with how to convert the term $c_3 (b_i \sum_{j=1}^{n} x_j)$ to the vector-matrix form. (and then factor for $x$ and get a solution for that.)

I appreciate any help and comment.

Answer 1

If $x,a,b$ are $n\times 1$ column vectors, then $$ \mathbf{1} \cdot x = \sum_{j = 1}^n x_j,$$ where $\mathbf{1}$ is a $1 \times n$ row vector of ones.

Thus, in the place of question marks, you need to fill $$c_3 \cdot b \cdot \mathbf{1} \cdot x. $$

You may check it for a small integer $n$, e.g. $n = 3$.