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Given ${3}\times{4}$ matrix $K$, ${4}\times{1}$ column vector $m$ and ${3}\times{1}$ column vector $p$, I have the following systems of equations in ${4}\times{4}$ homogeneous transformation matrix $T$

$$K T m = p$$

Apparently, matrix $T$ has $12$ unknown elements but an actual transformation needs only $6$ unknowns. Given several equations ($\geq 3$), can this equation be solved?

If so, I wonder which course or material provides with skills to rearrange and solve matrix equations like this one. Thank you!

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Since $T$ is a homogeneous transformation, it can be written as $$ T=\begin{bmatrix}\tilde{T}\\e_4^T\end{bmatrix}, $$ where $e_4=[0,0,0,1]^T$ and $\tilde{T}$ is $3\times 4$. Hence $p=KTm$ can be written as $$ p=[\tilde{K},k]\begin{bmatrix}\tilde{T}m\\e_4^Tm\end{bmatrix}=\tilde{K}\tilde{T}m+ke_4^Tm, $$ where $K=[\tilde{K},k]$ with $\tilde{K}$ being $3\times 3$ and $k$ the trailing column of $K$ ($3\times 1$). Hence we look for $\tilde{T}$ such that $$ \tilde{K} \tilde{T} m = p-ke_4^Tm $$ (note that $e_4^Tm$ is simply the last component of the vector $m$).

Next, we use the Kronecker product and the matrix vectorisation operator to transform (1) to system of linear equations of the form "matrix $\times$ vector = vector". Using this, we get that $$ \mathrm{vec}(\tilde{K}\tilde{T}m)=(m^T\otimes\tilde{K})\mathrm{vec}(\tilde{T}) $$ and hence (the entries of) the matrix $\tilde{T}$ can be found by solving the system $$ (m^T\otimes\tilde{K})\mathrm{vec}(\tilde{T})=p-ke_4^Tm, $$ where $m^T\otimes\tilde{K}$ is a $3\times 12$ matrix.

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  • $\begingroup$ @Pavel: sincerely thank you! how great this helps for me with little math background. I can almost find the references for all the clues $\endgroup$
    – Shawn Le
    Commented Mar 20, 2014 at 15:51

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