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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

  1. $AA^{T} = I$
  2. $\det(A) = 1$
  3. $\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?

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Informally speaking, the constraints are $a_i^T a_j = \delta_{ij}$, with $i \leq j$, so there are 6 constraints, with 9 variables, leaving 3 'independent' variables. –  copper.hat Jul 27 '12 at 20:00

5 Answers 5

up vote 2 down vote accepted

$A A^T = I$ already imposes $6$ independent constraints: $3$ saying that the rows should be orthogonal to each other, and $3$ saying that each row should be a vector of length $1$ (in particular the last three conditions you wrote down are redundant). These conditions also imply that $$\det(A) \det(A^T) = \det(A)^2 = 1$$

hence that $\det(A) = \pm 1$. The result we get is a $(9 - 6 = 3)$-dimensional manifold, namely the group $\text{O}(3)$ of rotations and reflections. The final condition $\det(A) = 1$ is an independent constraint but does not drop the dimension; it just singles out the connected component containing the rotations.

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Your matrices are elements of the Lie group $SO(3)$, which has $3$ free parameters, representing $2$ axis' and one rotation angle. This can be seen, when you look at the following:

Let $G$ be an element of $SO(3)$. Then $G=e^{g}$, with $g$ being an element of the Lie algebra $so(3)$, which is spanned by the traceless $3\times 3$ skew-symmetric matrices over $\Bbb{ R}$ $$ x = \left( \begin{array}{ccc} 0&1&0\\ -1&0&0\\ 0&0&0 \end{array}\right),\quad y = \left( \begin{array}{ccc} 0&0&0\\ 0&0&1\\ 0&-1&0 \end{array}\right),\quad z = \left( \begin{array}{ccc} 0&0&1\\ 0&0&0\\ -1&0&0 \end{array}\right). $$ which fulfill the commutation relation $[x,y]_-=xy-yx=z\;$. So $g=c_xx+c_yy+c_zz\;$ defines direction and angle ($|\vec c|^2$) of the rotation.

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So that means I should have 6 constraints. Do you know how I can express them? –  Hammer Jul 27 '12 at 20:03
    
You already did, as Qiaochu pointed out... –  draks ... Jul 27 '12 at 22:29

$SO(3)$ is a $3$-dimensional manifold, so there should be (locally) $6$ independent constraints.

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Do you know how I can express the 6 constraints? It seems even if I assume the above 5 I listed to be independent I am missing one –  Hammer Jul 27 '12 at 20:02
    
As @copper.hat noted in his comment, the 6 constraints can be the $(i,j)$ entries of $A^T A - I$ for $1 \le i \le j \le 3$. –  Robert Israel Jul 27 '12 at 20:17

The group of "rotation matrices" is usually called the special orthogonal group $\mathrm{SO}(3,\mathbb R)$. I assume that when you ask how many of the equations satisfied by these matrices are independent, you are asking for the smallest number of equations that all such matrices satisfy but no other (invertible) matrices satisfy. This is not quite rigorous, can be rigorously rephrased as the codimension of $\mathrm{SO}(3,\mathbb R)$ as a submanifold of $\mathrm{GL}(3,\mathbb R)$, the group of all $3\times 3$ invertible matrices. The dimension of $\mathrm{SO}(3,\mathbb R)$ as a real manifold is $3$, while that of $\mathrm{GL}(3,\mathbb R)$ is $9$, so the codimension is $6$.

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An interesting expression of a rotation matrix, found in Eringen - Microcontinuum field theories , is the following: $$ A_{ij}=\cos\phi\;\delta_{ij}-\sin\phi\;\varepsilon_{ijk}n_k+(1-\cos\phi)n_i n_j $$ where $\phi$ is the rotation angle and $\mathbf{n}$ is the rotation axis' unit vector.

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