Hi I am stuck at a problem , I know the answer but I need a proper mathematical way of writing it - How do I write the solution space of
$\underset{x}{\mathrm{arg min}}$$ \frac{1}{2}\|y-A_1\times A_2\times ....\times A_k \times x\|^2$
$A_1,A_2,....A_k$ are of any size and any rank.
I know it depends on the matrix with the least ranks but I need it proper mathematical defination , like we define for a vector space(with btw it will be in this case)
$V= \{x : x$ satisfy something $\}$
I am thinking answer should be
$W = \{ x: x = \{(A_1\times A_2\times ....\times A_k)^+\times y + v\}\}$ where $v\in V$ and $V$ is a vector space span by null space of $(A_1\times A_2\times ....\times A_k)$ and $(A_1\times A_2\times ....\times A_k)^+$ is pseudo inverse of $(A_1\times A_2\times ....\times A_k)$.
But I am not sure if this is the right way to write it , should I include $r$, where $r$ is the least rank of $A_i$ in $A_1,A_2,....A_k$ which is basically rank of $(A_1\times A_2\times ....\times A_k)$