How Sparse Can a Matrix with Orthogonal Rows Be?

Recently there has been interest in determining the combinatorial structure of orthogonal matrices (see, for example, [BBS, S, W]). This interest is largely motivated by Paul Halmo s' challenge at the ``Qualitative and Structured Matrix Theory'' conference in 1991 to characterize the sign-patterns of orthogonal matrices which have no zero entries, and Miroslav Fiedler's conjecture at the SIAM Applied Linear Algebra meeting in 1991 for the least number of nonzero entries in an n by n connected, orthogonal matrix. In this paper, which extends the work in [BBS, S], we study the question of how sparse a matrix with orthogonal rows can be under two natural notions of irreducibility. Define a real m by n matrix to be row-orthogonal if each of its rows is nonzero, and its rows are pairwise orthogonal. One may ask, what is the least number of nonzero entries in an m by n row-orthogonal matrix? Since Article ID jcta.1998.2898, available online at http: www.idealibrary.com on