# solving matrix equation by matrix rearranging

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!

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.