Solving a least square problems where the vectors are known and the unknown is the matrix

Question

I'm trying to solve a least-square problem of the form $Fc \approx d$ where $c$ and $d$ are known $3\times24$ matrices and $F$ is an unknown $3\times3$ matrix.

I was asked to try and solve a different problem in order to find $F$: reinterpreting $F$ as a $9\times1$ vector $x$ and solving $Ax = b$, but I have no idea what $A$ and $b$ should represent, only that they'd be found from $c$ and $d$.

Does anyone have an idea as to how I could do this?

Thanks in advance

Answer 1

Multiplying out $Fc$, and comparing entry-to-entry in $Fc\approx d$ gives a set of 96 equations in the nine unknown entries of $F$. A set of 96 equations in 9 unknowns can be formulated as $Ax\approx b$ for some known $96\times9$ matrix $A$ and some known $96\times 1$ matrix $b$.

There is no big mystery here, just shuffling equations around.

Answer 2

If we have the least-squares relation $\mathbf{Fc}=\mathbf d$ then $\mathbf c^T\mathbf F^T=\mathbf d^T$. Each column of $\mathbf F^T$ is then part of a separate least-squares problem (involving all of $\mathbf c^T$ and the corresponding column of $\mathbf d^T$), so $\mathbf F$ can be pieced together from the column-by-column results.