Given two full rank matrices $\mathbf{X}$ and $\mathbf{Y}$ of size $4 \times 4$. Each elements of the matrix are non-zero (randomly chosen). We can find a linear transformation matrix $\mathbf{A}$ such that $\mathbf{Y = AX}$. The solution will be $\mathbf{A = Y X^{-1}}$.
Now a constraint is added that the matrix $\mathbf{A}$ is block diagonal with two blocks and each block of size $2 \times 2$.
Also, the column vectors in $\mathbf{X}$ and $\mathbf{Y}$ can be chosen from subspaces of size $2$. (Each columns are chosen from different subspaces – That is eight subspaces in total). These subspaces are given say, $S_i$ for $i = 1,2,\ldots,8$. How to choose the columns vectors so that it is possible to find a block diagonal $\mathbf{A}$ matrix?
ps: some mathematical term or concept related to this or some geometrical interpretation of this would be of much help.
Thank you.
Best Answer
You have several questions, so I will answer one of them.
What happens if $X$ is the identity and $Y$ does not have blocks of size $2 \times 2$?