I have a $16$-dimensional real symmetric matrix with entries in $\{1,-1\}$.
$11$ of the rows are pairwise orthogonal, so are the remaining $5$ rows. But the two orthogonal sets are not necessarily pairwise orthogonal. In other words our matrix has two orthogonal sub-matrices, one $11$ by $16$ and the other $5$ by $16$.
What can we say about the maximum singular value (upper bound) of this matrix?