Suppose one has an $n \times n$ orthogonal matrix $M$:
$$
\left(
\begin{array}{ccc}
 0.239326 & 0.846726 &
 0.475161 \\
 0.768893 & 0.13356 &
 -0.625272 \\
 0.592897 & -0.514992 &
 0.619077 \\
\end{array}
\right)
$$
Because it is orthogonal, $M^T M = I$.
Suppose one entry of $M$ is erased, say $M(2,2)$:
$$
\left(
\begin{array}{ccc}
 0.239326 & 0.846726 &
 0.475161 \\
 0.768893 & \color{red}{x} &
 -0.625272 \\
 0.592897 & -0.514992 &
 0.619077 \\
\end{array}
\right)
$$
It can be recovered from $M^TM=I$. For example, we must have the $(2,2)$ entry 
of $M^TM$ equal to $1$:
\begin{eqnarray}
0.982162 + x^2 &=& 1 \\
x &=& \pm 0.13356
\end{eqnarray}
and then, e.g., entry $(2,1)$ of $M^TM$ disambiguates (or determines on its own):
\begin{eqnarray}
-0.102694 + 0.768893 x &=& 0 \\
x &=& 0.13356
\end{eqnarray}


My question is:



> 
> ***Q***. What is the maximum number $k$ of entries of an 
>  $n \times n$ orthogonal matrix $M$
>  that can be erased and then uniquely recovered, 
>  knowing only that $M$ is orthogonal?
> 
> 
> 


It could well be that $k$ depends on *which* entries are erased, which itself
could be interesting. But I am at the moment seeking the maximum of $k$ over
all possible entries that permits exact recovery.



How much redundancy resides in an $n \times n$ orthogonal matrix?
$O(n)$ is a manifold of dimension $n(n-1)/2$, so "generically" one might hope to recover up to $n(n+1)/2$ entries from the other $n(n-1)/2$. This won't quite work, however. Given any set $A$ of rows and any set $B$ of columns, generically we need at least one entry to survive that is either in both rows $A$ and columns $B$ or in neither: otherwise we could multiply each entry $a\_{ij}$ by $(-1)^{\chi\_A(i) + \chi\_B(j)}$ where $\chi\_A$ and $\chi\_B$ are the indicator functions of $A$ and $B$. Thus in the case $n=2$ we can only erase a single entry.


