The very simple, very formal description of the problem is this:
I have a real matrix $\mathrm{\textbf{Q}}$ and a real vector $\mathrm{\textbf{s}}$.
How could I calculate the vector $\mathrm{\textbf{w}}$, that I minimize $(\mathrm{\textbf{Q}}\mathrm{\textbf{w}}-\mathrm{\textbf{s}})^2$ for that?
$\mathrm{\textbf{Q}}$ is not surely square, thus $\mathrm{\textbf{w}}$ and $\mathrm{\textbf{s}}$ can have different dimensions. Singular cases are not a problem.
After some thinking I suspect these:
- I am essentially looking for the minimum of an n-dimensional quadratic potential. Thus, finding $\mathrm{\textbf{w}}$ if $\frac{d(\mathrm{\textbf{Q}}\mathrm{\textbf{w}}-\mathrm{\textbf{s}})^2}{\mathrm{d \textbf{w}}}=0$ will be likely a solution.
- The problem is likely to find the nearest points of two 1d lines in an n-dimensional space.
If it is true, then the solution is probably a simple formula, containing maybe a $\mathrm{\textbf{Q}}^{-1}$ and some elemental scalar vector operation. But as I tried to solve the problem, all my formulas became quickly chaotically unsolvable.