minimizing sum of different least squares?

Question

Can we write the minimization problem: $$\operatorname{min}\limits_{x\in\mathbb{R}^n}\sum_{i=1}^{n}\|C_i x-b_i\|_2^2$$ as a least square problem?

Answer 1

I'm going to assume that $C_i$ is a $m_i \times n$ matrix and $b_i$ is a $m_i \times 1$ column vector.

If you let $C = \begin{bmatrix}C_1 \\ C_2 \\ \vdots \\ C_n\end{bmatrix}$ and $b = \begin{bmatrix}b_1 \\ b_2 \\ \vdots \\ b_n\end{bmatrix}$, then the problem becomes $\displaystyle\min_{x \in \mathbb{R}^n}\|Cx-b\|_2^2$.

This is now a least squares problem.