Is there any method that can solve the matrix equation of the form $(AX)^2+(BY)^2=D$? $A$ and $B$ are matrices, $X$, $Y$ and $D$ are column vectors. (Solve for $X$ and $Y$)
I originally have two equations such as $AX=b$, $BY=c$. But $b^2+c^2=1$. So it goes to the single equation in the title, and makes $D=$1. It just means $b$ and $c$ have some relationship.
Please give me any suggestions you may have. Approximation, iteration, or any papers you can recommend. I would be so grateful for your help.
If there are any methods can solve using the form $AX=b$,$BY=c$ is also very good. But $b$ and $c$ are not known specifically.