Matrix rearrangement I have a matrix A in this form: 
$$
A=
\left[
\begin{array}{cccc}
x_1 & x_2 & 0 & 0 \\
0 & 0& x_1 & x_2 
\end{array} \right]
$$
 Here, $x_1$ and $x_2$ are variables, and A is a $2 \times 4$ matrix. I 
would like to rearrange the matrix product so that
$$
 A^TA=BC
$$
$A^T$ is the transpose of A. $B$ and $C$ are matrices to be found. $B$ is a $4 \times 2$ matrix that only contains constants, and C is a $2 \times 4$ matrix that contains constants and variables ($x_1,x_2$ etc). Is this possible to have such $B$ and $C$?

(Not really an answer but at least some development...)
Let's try it!
The left-hand side is easy to calculate:
$$
A^T A =
\left[
\begin{array}{cccc}
x_1^2 & x_1 x_2 & 0 & 0 \\
x_1 x_2 & x_2^2 & 0 & 0 \\
0 & 0 &x_1^2 & x_1 x_2  \\
0 & 0 &x_1 x_2 & x_2^2  \\
\end{array}
\right]
$$
The right-hand side should have this form also. Let's see what happens if we set
$$
B_0 = 
\left[
\begin{array}{cc}
b_{00} & b_{01} \\
b_{10} & b_{11} \\
\end{array}
\right]
\qquad
B_1 = 
\left[
\begin{array}{cc}
b_{20} & b_{21} \\
b_{30} & b_{31} \\
\end{array}
\right] \qquad
\Rightarrow \qquad
B_1 = 
\left[
\begin{array}{c}
B_0 \\ B_1
\end{array}
\right]
$$
and 
$$
C_0=
\left[
\begin{array}{cc}
c_{00} & c_{01} & \\
c_{10} & c_{11} & \\
\end{array}
\right]\qquad 
C_1=
\left[
\begin{array}{cc}
c_{02} & c_{03} \\
 c_{12} & c_{13} \\
\end{array}
\right]\qquad
\Rightarrow \qquad
C=
\left[
\begin{array}{cc} C_0 & C_1 \\
\end{array}
\right]
$$
Now their product is
$$
BC = \left[
\begin{array}{cc}
B_0 C_0 & B_0 C_1 \\
B_1 C_0 & B_1 C_1
\end{array}
\right]
$$
Now we get the equations
$$
\begin{array}{cl}
B_0 C_1 = B_1 C_0 &= 
\left[
\begin{array}{cc}
0& 0 \\
0 & 0 \\
\end{array}
\right] \\
B_0 C_0 = B_1C_1
&= 
\left[
\begin{array}{cc}
x_1^2 & x_1 x_ 2 \\
x_1 x_2 & x_2^2 \\
\end{array}
\right] \\
\end{array}
$$
Now, some insight would be needed to continue ...
