THE EXISTENCE THEOREM OF ORTHOGONAL MATRICES WITH p NONZERO ENTRIES

It was shown that if Q is a fully indecomposable $n{\times}n$ orthogonal matrix then Q has at least 4n-4 nonzero entries in 1993. In this paper, we show that for each integer p with $4n-4{\leq}p{\leq}n^2$, there exist a fully indecomposable $n{\times}n$ orthogonal matrix with exactly p nonzero entries. Furthermore, we obtain a method of construction of a fully indecomposable $n{\times}n$ orthogonal matrix which has exactly 4n-4 nonzero entries. This is a part of the study in sparse matrices.