Is 5 entries for an orthogonal matrix enough for reconstruction?

Question

Let
$R=\begin{bmatrix} r_{11} & r_{12} & r_{13} \\ u_{22} & u_{23} & r_{23} \\ u_{31} & u_{32} & r_{33} \end{bmatrix}$ be orthogonal $3 \times{3}$ matrix where entries $r_{ij}$ are given

and $u_{ij}$ are unknown.

Question for a general case:

• is it possible to reconstruct matrix $R$ in an unique way i.e. there is only one possible set of values $u_{ij}$ for orthogonal matrix $R$ with the known $r_{ij}$ entries ?

Example:

$R=\begin{bmatrix} -\dfrac{1}{3} & \dfrac{2}{3} & \dfrac{2}{3} \\ * & * & \dfrac{2}{3} \\ * & * & -\dfrac{1}{3} \\ \end{bmatrix} $

Answer 1

I would reason as follows. The second column, thought of as a vector, must be orthogonal to the third column, which means it lies on a certain plane; we know its first entry, which restricts it to a line in that plane; and we know its norm, which restricts it to at most two points on that line. Then the first column must be the cross product of the other two columns (or the negative of the cross product). So in general, there won't be many solutions; and, if you think about what could go wrong with the argument (as in the example Vincent gives), you'll probably be able to work out whether there are any more counterexamples.

Answer 2

Not in general: take the case where $r_{13} = 1$ and the other $r$'s are $0$. Then the possibilities for the $u_{ij}$ are somewhat restricted (they need to be of the form $\cos \theta, -\sin \theta, \sin \theta, \cos \theta$) but not uniquely determined.