I have given two matrices $A,N$ and two vectors $a,b$ with $A,N \in \mathbb{C}^{n \ast m}$ and $a, b \in \mathbb{C}^{n,m}$ where $A_{n,m} = a_n b_m + N_{n,m}$.
$N$ models white noise (gaussian distributed).
Given $A$ and $b_1 = 1$, how can I calculate the best estimation of $a$ and $b$?
(I already thought of LU decomposition and only use the first column of L and the first row of U as an initial estimation, but I doubt, that's the best possible estimation.)
[Edit 21.12.2015 21:49 UTC]
The best estimation of $a$ and $b$ is considered this one, having a minimal value of $Q := \sum_{j=1}^{m} {|| A_{j} - a b_j ||}^2$, where $|| \circ ||$ denotes the euclidian norm and $A_{j}$ is the $j$th column of $A$.