I am trying to solve a system of non-linear equations. I know that 9 of my variables put together form a 3x3 rotation matrix
$ A = \left( \begin{matrix} a_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33} \end{matrix} \right) $
There are many properties of rotation matrices for example
- $AA^{T} = I$
- $\det(A) = 1$
- $\mathrm{magnitude}(a_{11},a_{21},a_{31}) = 1\\ \mathrm{magnitude} (a_{12},a_{22},a_{32}) = 1\\ \mathrm{magnitude}(a_{13},a_{23},a_{33}) = 1 $
I'm sure some of you could provide me with various others. My question is how many of them are independent, and which ones are independent? If these are 9 unknowns, how many useful equations does this constraint give me?