ZPC matrices and zero cycles (Q606420)

Summary: Let \(H\) be an \(m \times n\) real matrix and let \(Z_i\) be the set of column indices of the zero entries of row \(i\) of \(H\). Then the conditions \(|Z_k \cap \left(\bigcup_{i=1}^{k-1} Z_i \right)| \leq 1\) for all \(k\) (\(2 \leq k \leq m\)) are called the (row) Zero Position Conditions (ZPCs). If \(H\) satisfies the ZPC, then \(H\) is said to be a (row) ZPC matrix. If \(H^T\) satisfies the ZPC, then \(H\) is said to be a column ZPC matrix. The real matrix \(H\) is said to have a zero cycle if \(H\) has a sequence of at least four zero entries of the form \(h_{i_1j_1}, h_{i_1j_2},h_{i_2j_2},h_{i_2j_3},\dots,h_{i_kj_k},h_{i_kj_1}\) in which the consecutive entries alternatively share the same row or column index (but not both), and the last entry has one common index with the first entry. Several connections between the ZPC and the nonexistence of zero cycles are established. In particular, it is proved that a matrix \(H\) has no zero cycle if and only if there are permutation matrices \(P\) and \(Q\) such that PHQ is a row ZPC matrix and a column ZPC matrix.