I'm trying to obtain original transform matrix $A$ and its translation vector $b$ From $y=Ax+b$ equation. I have original values of vectors before transform and translation $(x)$ and vectors after transform and translation $(y)$
I know that finding two unknowns with $1$ equation is not possible, but I think if I arrange $y=Ax+b$ in such a way that I can obtain $A+b$ or $A$ only? Is there is a way to find out?
And if I drop $b$ from equation and try to find out matrix $A$ only $y = Ax$ when vectors $y$ and $x$ $(2\times1)$ are known and matrix $A$ $(2\times2)$ is unknown, how I can do that? Because I can't take inverse of $x$
We have $$\begin{align*} \underset{2 \times 1}{y} = \underset{2 \times 2}{A}\times\underset{2 \times 1}{x} \end{align*}$$ Let $\underset{1 \times 2}{x^+}$ be the Moore-Penrose inverse of $x$ defined by $$x^+ =\dfrac{x^\mathrm{T}}{\langle x,x\rangle}=\dfrac{x^\mathrm{T}}{x x^\mathrm{T}}$$ Then we get that $$ \begin{align*} A = yx^+ \end{align*}$$