Let $Y_{N\times1} = A_{N \times N}X_{N \times M}W_{M \times 1} + N_{N\times 1}$, where $A$ is a diagonal matrix with the elements $a_{i,i}$ being a function of array index ($a_{i,i} = k_0k^i$, where $i$ is the row/column index of the diagonal matrix $A$ and both $k_0$ and $k$ are uknown scalar values), $X$ is the input matrix and $W$ is the coefficient vector and $N$ is the noise vector, with a $0$ mean unit variance Gaussian pdf.
$Y$ is the output vector. Given $Y$ and $X$ are known, is there a way to solve for both $A$ and $W$, provided N > M+1 ?
For a general set of linear equations $Y = XW + N$, I can solve using Least Squares, as long as M <=N, but I am not sure how to solve the above, where we have an additional coefficient matrix $A$, (although it adds just 1 extra unknown, i.e. '$k$'). Thanks!
The problem is not well determined. Evidently you have in mind some linear regression procedure similar to how one might estimate the solution to the seemingly simpler equation:
$$ Y = XW + N $$
with the interpretation that "the noise vector" N will be minimized.
Your altered equation:
$$ Y = AXW + N $$
introduces a diagonal matrix $A$ whose diagonal entries form a geometric series. Various statements about the parameterization of $A$ were made, but in each case we have $A$ invertible for any nonzero choice for parameter $k$.
Notice that if $k\neq 0$, multiplying both sides on the left by $A^{-1}$, it becomes:
$$ A^{-1} Y = XW + A^{-1}N $$
Now one can apply the linear regression you proposed for the simpler problem setup, solving for matrix $W$ and noise vector $N$. In other words you can apply whatever choices for $A$ you prefer (so long as it is invertible), and settle for solutions $W,N$ provided by the earlier procedure.
Of course one might object that this does not minimize the objective function you intended for $N$, but repeated requests for clarification did not elicit any information about that criteria or the underlying probability model.
Think carefully about what motivates your problem. I'm convinced you have a real motive (application) but have not thought through what essential characteristics would determine the "best" $A$.