This may be a simple question, but I couldn't find a solution because I didn't know what to search for. Excuse my math.
I have a linear system $AX = Y$. $A$ is an affine transform and all the coefficients are known. $Y$ is the solution, and I know them. I need to find $X$. The reason I do this is because I need to find the magnitude of column vectors in $Y$ that corresponds to those in $X$. Dimensionality of these matrices are $A \in \mathbb{R}^{d \times 512}, X \in \mathbb{R}^ {512 \times k}, Y \in \mathbb{R}^{d \times k} $. $k$ is the number of column vectors, and $d$ is the dimensionality of output space. $A$ is not invertible.
Also, is there a way to solve this using least-squares? I don't think I can, but it was suggested that I use least-squares from someone.
Thanks!
If you want to solve $Ax = y$ in terms of least squares (more precisely, $\arg\min_x \|Ax - y\|$), then you can write $x = A^+ y$, where $A^+$ is pseudoinverse. In your matrix case, you essentially have $k$ independent systems, and solution for them can be written in the same form simultaneously: $X = A^+ Y$