Patterns that preserve sparsity in orthogonal factorization

An m × n zero-nonzero pattern with the Hall property allows a full rank matrix A ε with a QR factorization. The union of patterns occurring in Q over all such A is denoted by . By further restricting to have the strong Hall property, a Hasse diagram that is a forest is used to characterize patterns that yield = , thus preserving the sparsity of . For fixed n, the sparsest n × n such patterns are characterized by a binary rooted tree.