Convert Least Square equations to matrix form I want to minimize following equation using Least Square:
$$
f = min\sum_{j=1}^{m} \sum_{i=1}^{n}(p_{ij}a_{i} + q_{ij}b_{i} + r_{ij}tx_{j} + s_{ij}ty_{j})^{2}
$$
with respect to $a_{i}$, $b_{i}$, $tx_{j}$, $ty_{j}$. 
And $p_{ij}$,$q_{ij}$,$r_{ij}$, $s_{ij}$ are known.
What I did is to take the partial derivative of f with respect to $a_{i}$, $b_{i}$, $tx_{j}$, $ty_{j}$ respectively and set those partial derivative to $0$ to get all the equations.
For example, for partial derivate of f wrt $a_{i}$:
$$
\sum_{j=1}^{m} \sum_{i=1}^{n}2(p_{ij}a_{i} + q_{ij}b_{i} + r_{ij}tx_{j} + s_{ij}ty_{j})p_{ij} = 0
$$
My question is how can I convert all the equations I got from those partial derivatives and convert them to a Matrix form to solve for all the $a_{i}$, $b_{i}$, $tx_{j}$, $ty_{j}$ parameters.
For example, the final matrix form should be some Matrix A multiply vector X,
$$
AX=0
$$
where X:
$$
X=\begin{bmatrix}
a_{1}\\ 
a_{2}\\ 
.\\ 
.\\ 
.\\ 
a_{n}\\ 
b_{1}\\ 
b_{2}\\ 
.\\ 
.\\ 
.\\ 
b_{n}\\ 
tx_{1}\\ 
tx_{2}\\ 
.\\ 
.\\ 
.\\ 
tx_{m}\\ 
ty_{1}\\ 
ty_{2}\\ 
.\\ 
.\\ 
.\\ 
ty_{m}\\ 
\end{bmatrix}
$$
How can I get A?
Thanks very much.
 A: Problem statement
Given four matrices
$$
\mathbf{P}, \mathbf{Q}, \mathbf{R}, \mathbf{S} \in \mathbb{C}^{m\times n},
$$
Find the four solution vectors
$$
 a, b, x, y \in \mathbb{C}^{n}
$$
which solve the equation
$$
 \mathbf{P} a + \mathbf{Q} b + \mathbf{R} x + \mathbf{S} y = 0.
$$
By assumption, there is no exact solution.
The observation of @littleO is to create a block formulation:
$$
\left[ \begin{array}{cccc}
  \mathbf{P} & \mathbf{Q} & \mathbf{R} & \mathbf{S}
\end{array} \right]
%
\left[ \begin{array}{c}
  a \\ b \\ x \\ y
\end{array} \right]
=
\mathbf{O}
$$
where $\mathbf{0}$ has dimension $4n\times 1$.
The problem is now one of reduction to find the null space.
The method of least squares 
A careful statement of the least squares problem starts with the linear system
$$
\mathbf{A} x = b
$$
with
$$
\mathbf{A}\in\mathbb{C}^{m\times n},\, x \in \mathbb{C}^{n},\, b \in \mathbb{C}^{m}.
$$
The problem provides the matrix $\mathbf{A}$, the data vector $b$, and asks for the solution vector $x$. 
In this case
$$
 \mathbf{A} = \left[ \begin{array}{cccc}
  \mathbf{P} & \mathbf{Q} & \mathbf{R} & \mathbf{S}
\end{array} \right]
$$
And the matrix maps data in measurement space, an $n-$space, onto an affine parameter space, an $m-$space. The goal is to find the least squares solution defined as
$$
 x_{LS} = \left\{
x \in \mathbb{C}^{n} \colon 
\lVert
  \mathbf{A} x - b
\rVert_{2}^{2}
\text{ is minimized}
\right\}
$$
The full statement of the least squares requirement is that the data vector is not in the null space:
$$
 b \notin \mathcal{N}\left( \mathbf{A} \right)
$$
This demand insures existence of a solution.
Changing the problem
To conclude: as written we do not have a least squares problem, we have the challenge of finding a span for the null space of the block matrix $\mathbf{A}$. If you were to stir in a nonzero $m-$vector, say, $\beta \notin \mathcal{N}\left( \mathbf{A} \right)$, we would have a least squares problem:
$$
\left[ \begin{array}{cccc}
  \mathbf{P} & \mathbf{Q} & \mathbf{R} & \mathbf{S}
\end{array} \right]
%
\left[ \begin{array}{c}
  a \\ b \\ x \\ y
\end{array} \right]
=
\beta
$$
A: Building on what @dantopia wrote, the problem is that the RHS of
$$
\left[ \begin{array}{cccc}
  \mathbf{P} & \mathbf{Q} & \mathbf{R} & \mathbf{S}
\end{array} \right]
%
\left[ \begin{array}{c}
  a \\ b \\ x \\ y
\end{array} \right]
=
\mathbf{O}
$$
is 0, or in short form 
$$
\mathbf{X}\boldsymbol{\beta}=\mathbf{y}\,. $$
 Wouldn't the resolution to the problem be to define a set of related variables which would simply yield a non-zero RHS, like this 
$$
\boldsymbol{\beta}' \equiv \left[ \begin{array}{c}
  a' \\ b' \\ x' \\ y'
\end{array} \right]
\equiv
\left[ \begin{array}{c}
  a+1 \\ b+1 \\ x +1 \\ y +1
\end{array} \right] = \boldsymbol{\beta} + \boldsymbol{1}
$$
I've picked a vector of 1s here, but it could be any constant offset that is known and doesn't make the RHS 0. Then the top equation becomes 
$$
\left[ \begin{array}{cccc}
  \mathbf{P} & \mathbf{Q} & \mathbf{R} & \mathbf{S}
\end{array} \right]
%
\left[ \begin{array}{c}
  a \\ b \\ x \\ y
\end{array} \right]
=
\left[ \begin{array}{c}
  \mathbf{P}\cdot 1 \\ \mathbf{Q}\cdot 1 \\ \mathbf{R} \cdot 1\\  \mathbf{S} \cdot 1
\end{array} \right]
$$
Thus we would solve for the primed quantities in the least squares, and then back out the desired un-primmed ones from this equation. 
The equation to minimize then becomes in the standard least squares  form 
$$
\mathbf{X}\boldsymbol{\beta}'=\mathbf{y}\,.
$$
This allows us to employ the standard method of least squares to an overdetermined system of equations
$$
\hat{\boldsymbol{\beta}}' = (\mathbf{X}^{{\rm T}}\mathbf{X})^{-1}\mathbf{X}^{{\rm T}}\mathbf{y}\,. $$
Finally, your solution is 
$$
\hat{\boldsymbol{\beta}} = \hat{\boldsymbol{\beta}}' - \boldsymbol{1}
$$ 
