Algorithm concerning orthogonal matrices Say I have a n-dimensional orthogonal matrix, with some of its elements given and these others unknown. Does there exist an effective algorithm to find out the unknown elements and restore the whole matrix (it only needs to find one solution if there are many, and gives an error when no solution exists)?
Thanks!

A matrix $A$ is orthogonal iff $$A^T A = I.$$ If we denote the (known or unknown) $(i, j)$ entry as usual by $a_{ij}$ then the above condition is equivalent to the system
$$\sum_{k = 1}^n a_{ik} a_{jk} = \delta_{ij}, \qquad 1 \leq i \leq j \leq n,$$
where $\delta_{ij}$ is the Kronecker delta symbol (that is, it has value $1$ if $i = j$ and value $0$ otherwise). This reduces the problem to solving a (necessarily) quadratic system of $\frac{1}{2}n (n + 1)$ equations in the unknown entries $a_{ij}$.
