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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.

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    Apparently I'm missing something. If you want $\mathbf{Y = AX}$ and $\mathbf Y$ and $\mathbf X$ are already determined, then it's not very likely for $\mathbf A$ to be block-diagonal... what's your *actual* problem?2011-09-13
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    Thanks. you are right. i will modify the problem and clearly state it.2011-09-13

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