I'm struggling to solve the following equation
$$ R^{*}*t = (||t||\ 0\ 0)^{T} $$ where R* is a 3x3 matrix and t a 3-vector. On the right side I have a column vector with t's frobenius norm and 2 zeroes.
I re-wrote the equation like this but that didn't get me any closer to solving it for R*. So I was wndering if it is even possible? Or if I just made a basic mistake.
$$ R_{11}t_{1}+R_{12}t_{2}+R_{13}t_{3} = ||t||\\ R_{21}t_{1}+R_{22}t_{2}+R_{23}t_{3} = 0\\ R_{31}t_{1}+R_{32}t_{2}+R_{33}t_{3} = 0\\ $$
Thanks for your help!
I believe you can follow this method to see if the equations are solvable. In this case, your matrix $M$ will be:
$$ M= \left[ {\begin{array}{cc} R_{11} & R_{12} & R_{13} & | & ||t|| \\ R_{21} & R_{22} & R_{23} & | & 0 \\ R_{31} & R_{32} & R_{33} & | & 0\\ \end{array} } \right] $$
which is actually a shorthand for:
$$ M= \left[ {\begin{array}{cc} R_{11} & R_{12} & R_{13}\\ R_{21} & R_{22} & R_{23}\\ R_{31} & R_{32} & R_{33}\\ \end{array} } \right] \left[ {\begin{array}{cc} t_{1}\\ t_{2}\\ t_{3}\\ \end{array} } \right] = \left[ {\begin{array}{cc} ||t||\\ 0\\ 0\\ \end{array} } \right] $$
See if you can simplify the $3 \times 3$ matrix to become the identity matrix $I$.