There is a 7-by-6 matrix $H$ given. Its rank is 6. I'd like to design a 6-by-5 matrix $D$ such that the following holds:
$ \left[ \begin{array}{l} l_1(a_1, a_2, a_3, a_4) \\ l_2(a_1, a_2, a_3, a_4) \\ l_3(a_1, a_2, a_3, a_4) \\ l_4(a_1, a_2, a_3, a_4) \\ l_5(l_1, a_5) \\ l_6(l_1, a_5) \\ l_7(l_1, a_5) \end{array} \right] $ = $H$$D$ $\left[ \begin{array}{l} a_1 \\ a_2 \\ a_3 \\ a_4 \\ a_5 \end{array} \right]$,
where $l_k(a,b,c)$ denotes a linear combination of $a, b, c$, and subscript $k$ is to denote different linear combinations.
The top four linear combinations of $a_1, a_2, a_3, a_4$ are independent. Two out of the bottom three linear combinations of $l_1, a_5$ are independent.
This probem arises from a communication problem where a transmitter has multiple antennas, and the total number of antennas of multiple receivers is greater that the number of antennas the transmitter has. The transmitter wishes to transmit different symbols to different receivers.
If my description is not clear enough, please ask. I need help from who are familiar with matrix decomposition, linear transformation, etc.
Thanks.
Let $L$ be the $7 \times 5$ matrix such that $L\begin{bmatrix}a_1\\a_2\\a_3\\a_4\\a_5\end{bmatrix} = \begin{bmatrix}l_1(a_1, a_2, a_3, a_4) \\l_2(a_1, a_2, a_3, a_4) \\l_3(a_1, a_2, a_3, a_4) \\l_4(a_1, a_2, a_3, a_4) \\l_5(l_1, a_5) \\l_6(l_1, a_5) \\l_7(l_1, a_5)\end{bmatrix}$.
We wish to find a $6 \times 5$ matrix $D$ such that $HD\vec{a} = L\vec{a}$ for every $\vec{a}$, i.e. $HD = L$.
A solution (if one exists) is $D = H^{+}L$ where the $H^{+}$ is the pseudoinverse of $H$.
Since $H$ is a $7 \times 6$ matrix with rank $6$, $H$ has linearly independent columns.
Hence, $H^{+} = (H^TH)^{-1}H^T$, and thus, $D = (H^TH)^{-1}H^TL$.