I have a set of RGB colour values detected by a camera $C_{i_{RGB}}$ which are to be described by the following:
$C_{i_{RGB}} = X F_{i_{rgb}}$
where $F_{i_{rgb}}$ is the component incident light at the detector, where the components $rgb$ are fixed at narrow bands and $X$, an $(n\times 3)$ matrix, is a description of the colour band overlap (cross talk if you will) for specific frequencies. In a simplified RGB scheme, $X$ would be described:
$\begin{align} \mathsf X &= \begin{bmatrix} 1 & R_{GR} & R_{BR} \\ R_{RG} & 1 & R_{BG} \\ R_{RB} & R_{G} & 1 \end{bmatrix}\end{align}$
That is, the diagonal entries of $X$ are $1$'s, and the off-diagonals are rational.
How can I solve for the values of $X$ only knowing $C_{i_{RGB}}$? Eventually, given $C_i$ I need to find the corresponding $F_i$ using the constant $X$.
Eventually added colour bands (band passes) will be added which will turn the above relation into something like this (and I fear the case where I have 5, 6, and 7 band passes to deal with):
$\begin{bmatrix} C_{i_R} \\ C_{i_G} \\ C_{i_B} \end{bmatrix} = \begin{align}\begin{bmatrix} X_{aR} & X_{bR} & X_{cR} & X_{dR} & X_{eR} & X_{fR} \\ X_{aG} & X_{bG} & X_{cG} & X_{dG} & X_{eG} & X_{fG} \\ X_{aB} & X_{bB} & X_{cB} & X_{dB} & X_{eB} & X_{fB} \end{bmatrix}\end{align}\begin{bmatrix} F_{i_a} \\ F_{i_b} \\ F_{i_c} \\ F_{i_d} \\ F_{i_e} \\ F_{i_f} \end{bmatrix}$
This is for a graphics programming application and I lament that my first year algebra was 20 years ago.